HOW THE BODY CONTRIBUTES TO THE WAKE IN UNDULATORY FISH SWIMMING : FLOW FIELDS OF A SWIMMING EEL (ANGUILLA ANGUILLA)
Department of Marine Biology, University of Groningen, PO Box 14,
9750 AA Haren (Gn), The Netherlands
*
Present address and address for correspondence: Department of Zoology,
University of Cambridge, Downing Street, Cambridge CB2 3EJ, UK (e-mail:
ukm20{at}cam.ac.uk
)
Accepted May 31, 2001
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Summary |
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Key words: fish, swimming, undulatory swimming, eel, Anguilla anguilla, flow visualisation, particle image velocimetry
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Introduction |
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Simple analytical models predict that undulatory swimmers shed a reverse
von Kármán
vortex street when optimising swimming performance (Lighthill,
1969). This wake consists of a
double row of single vortices and is generated if the fish swims at a slip
(the ratio of swimming speed U to body wave speed V) of less
than 1. Typically, slip is in the region of 0.6-0.8 (for a review, see
Videler, 1993
). Computational
simulations of the flow around an undulatory swimmer exist for a danio
(Wolfgang et al., 1999
) and
eel (Carling et al., 1998
;
Pedley and Hill, 1999
). Two of
the three models predict a wake resembling the prediction of Lighthill
(Lighthill, 1969
) with one
vortex shed per half tail-beat. Carling et al. (Carling et al.,
1998
) predict a wake consisting
of two counter-rotating vortices on either side of the mean path of motion,
into which all shed vortices merge. However, model studies on waving plates
have shown that the wake changes significantly with slip (Hertel,
1966
) and the phase between
interacting waving plates (Gopalkrishnan et al.,
1994
; Streitlien et al.,
1996
).
The experimental flow fields published so far have concentrated on the wake
rather than on wake generation. Two main wake patterns have been observed
behind undulatory swimmers (Fig.
1). The reverse von
Kármán vortex
street, indicative of high efficiency, has been observed in eel (Gray,
1968), bream Diplodus
annularis (Aleyev, 1977
),
trout Oncorhynchus mykiss (Blickhan et al.,
1992
) and mullet
(Müller et al.,
1997
). A different vortex
pattern resembling a double row of double vortices has been reported for zebra
danio Brachydanio albolineatus (Rosen,
1959
), water snake Natrix
natrix (Hertel, 1966
) and
Kuhli leach Acanthophthalmus kuhli (Rayner,
1995
). All these wake patterns
were generated during steady undulatory swimming at speeds between 1 and
7Ls-1, where L is body length. The values for
slip were 0.6-0.7 where mentioned.
|
In the present study, we address the question of how a wake is generated by
the interaction between fish and water. Quantitative mapping of the flow, from
its generation on the body to the shedding of the wake, will provide a time
course of wake development. If thrust is generated along the whole body, flow
speeds adjacent to the body should increase continuously from head to tail,
rather than increase sharply towards the tail as observed in carangiform
swimmers (Müller et al.,
1997; Wolfgang et al.,
1999
). Mapping swimming
movements and flow simultaneously will suggest the relevant kinematic
parameters for thrust production. Key wave parameters such as maximum lateral
displacement and lateral velocity will be mapped relative to the flow to
establish correlations between body wave and flow. Previous work has shown
that eels generate flow fields around their body and in their wake that are
qualitatively equivalent to the flow fields of carangiform swimmers such as
mullet and danio. The eel is also the most extreme case of body-generated
thrust: without a pronounced tail, an eel generates thrust along the whole of
its body. Hence, eels provide the ideal showcase for thrust-generating
mechanisms of the body during steady undulatory swimming.
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Materials and methods |
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Flow visualisation
The experiments were performed in still-standing water. This allows the eel
to display its preferred swimming behaviours spontaneously. In still-standing
water, the signal-to-noise ratio in the flow visualisation is maximal in the
absence of turbulence induced by the flow tank. The eels swam in a 0.2
mx0.3 mx0.3 m tank filled to a depth of at least 50 mm with
artificial sea water (salinity 30, temperature 22 °C). The water
was seeded with unexpanded polystyrene particles (VF 654, BASF, diameter
0.2-0.4 mm). The particles were slightly denser (1.026 kg m-3) than
sea water, so their response to changes in the flow velocity was delayed
(Merzkirch, 1987
) and they
sank slowly. Both effects were insignificant compared with the fish-generated
flows and were therefore neglected. The particles were illuminated in a
horizontal plane by a 1 mm thin laser light sheet (krypton ion laser,
wavelength
=647 nm, maximum power 0.8 W). A horizontal light sheet was
positioned in the middle of the water column. The water surface was covered
with a Plexiglas raft to prevent surface waves from distorting the images when
the fish was swimming too close to the water surface. We recorded sequences
only when the fish was swimming in the middle of the tank to avoid wall and
surface effects. Only spontaneous swimming behaviour was recorded; we did not
stimulate the fish in any way.
All experiments were performed with a single fish in the test tank. The
fish avoided looking into the light sheet by keeping its eyes above or below
the level of the light sheet. Otherwise, swimming behaviour seemed to be
unaffected by the light sheet. A CCD camera (Adimec MX-12, with 50 mm lens and
5 mm extension ring) was mounted perpendicular to the light sheet to record
top-view images of 1024x1024 pixels at a frequency of 25 images
s-1 (integration time 10 ms). Recordings of the flow fields were
made using a purpose-designed recording system (Dutch Vision Systems)
(Müller et al.,
1997). The recorded images
were checked immediately, and sequences in which the eel and the wake were in
the light sheet in the centre of the field of view were stored uncompressed as
512x512 pixel images for later analysis. We filmed in the centre of the
tank to avoid recordings impaired by wall effects. We recorded up to five
sequences from 11 eels.
Kinematic analysis
In the selected sequences, the eel was swimming horizontally through the
light sheet along a straight path through the centre of the field of view,
i.e. with its body at least 25 mm away from the water surface and the bottom
of the tank. We obtained the midlines of the swimming eel from the digitised
images (TIM, Dutch Vision Systems) (Müller et
al., 1997). The body length
was assumed to be equivalent to the number of pixels representing the midline
of the fish. The instantaneous swimming speed was obtained from the head
position in sequential images and the frame rate. It was averaged over
complete tail-beat cycles to obtain the mean swimming speed U. The
mean path of motion was calculated from the head position in consecutive
images using standard linear regression over complete tail-beat cycles.
In an earth-bound frame of reference, we calculated the following kinematic
parameters. The amplitude A of the body wave at each point along the
body was defined as half the transverse distance between the points of maximum
lateral excursion of a particular body segment. The stride length
s of the fish was calculated as twice the distance between
two consecutive points where the path of the tail crosses the mean path of
motion. We also determined the lateral velocity V(t,x) of
the body relative to the fluid and the points on the midline where this
lateral velocity is zero [for a definition of V(t,x), see
Appendix].
The following kinematic parameters were obtained in a fishbound frame of
reference. The body wave length b was considered to be
twice the distance between two consecutive points where the body midline
crosses the mean path of motion. The speed V of wave propagation was
calculated from the displacement of these crossings and the frame rate (25
Hz). We further determined the position of the nodes, maxima and inflection
points of the midline. The nodes were defined as the points where the midline
crosses the mean path of motion. The inflection points were defined as the
points where the curvature of the midline changes direction. Nodes, maxima and
inflection points of the midline were derived numerically by searching for
zero positions and extremes in the lateral displacement function and its
derivatives (Müller et al.,
1997
).
Hydrodynamic analysis
Particle image velocimetry
The flow generated by the swimming fish was visualised using
two-dimensional particle image velocimetry (PIV). The exact algorithms
employed to derive the flow fields from our recordings are described elsewhere
(Stamhuis and Videler,
1995).
We conducted subimage cross-correlations on pairs of consecutive images to
obtain a velocity vector field from the particle displacements (Chen et al.,
1993; Stamhuis and Videler,
1995
). We used a subimage size
of 33x33 pixels with 50 % overlap. The choice of the subimage size was
determined by the seeding density of the polystyrene particles and the peak
velocities in the flow fields (Keane and Adrian,
1991
): there were more than 10
particles in each subimage; the particle displacement in the flow was less
than 25 % of the subimage diameter between consecutive frames. The
cross-correlation resulted in vector fields of, at best, 30x30 equally
spaced vectors if all subimages contained sufficient information for
cross-correlation. A comparison of particle tracking results with PIV flow
velocity data for the same area in the flow field revealed no significant
underestimation of the peak flow velocities due to the spatial averaging of
the cross-correlation. Regions in which PIV was not possible or was unreliable
were augmented by particle tracking velocimetry (as described in
Müller et al.,
1997
).
Post-processing
The flow velocity vectors resulting from PIV were fitted into a grid of
30x30 cells. Gaps in the resulting vector field were filled using a
two-dimensional spline to interpolate vectors (Stamhuis and Videler,
1995). The following flow
parameters were derived from the flow field: vorticity
, which is
proportional to the angular velocity at a point in the fluid, and the
discriminant for complex eigenvalues d, which was used to locate the
centre of vortices in the wake (Vollmers et al.,
1983
; Stamhuis and Videler,
1995
).
The wake was characterised by the position of the vortex centres and the
direction of the jet. From the position of the vortex centres in the flow
field, we calculated the distance between vortices along and perpendicular to
the mean path of motion. Consecutive vortices were considered to be a pair
and, hence, were assumed to form a vortex ring in the three-dimensional wake
if they were separated by a distinct jet flow and moved away from the mean
path of motion together. We also confirmed that their respective circulations
were equal in magnitude and of opposite sense after shedding. The forward
momentum that the fish is able to gain from such a vortex ring depends on the
momentum angle between the ring and the mean path of motion. The angle
between the ring and the mean path of motion and the flow speed of the jet
through the ring were obtained directly from the velocity vector field.
All mean values in the text are given ± 1 S.D.
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Results |
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Swimming kinematics
In the selected three sequences (Fig.
2), the eels have similar tail-beat frequencies, amplitudes and
body wave lengths (Table 1). They cross the field of view within three tail-beat cycles. Their slip
U/V ranges from 0.6 to 0.7 at swimming speeds of 0.10-0.14 m
s-1 (1.0-1.5Ls-1). Within one tail-beat cycle,
the swimming speed varies between 0.9 and 1.1U. Swimming speed
exhibits a total of two peaks per half tail-beat cycle, with both peaks
clustering around the moment of stroke reversal of the tail. The kinematic
characteristics of the undulating body movements agree with earlier findings
(Hess, 1983; Videler,
1993
). The amplitude envelope
of the body wave shows a minimum at a distance 0.1L from the snout
tip and increases from there almost linearly to a maximum amplitude of
approximately 0.1L at the tail tip. The body wave length decreases
along the body by 20-30 % from above 1.0L in the first third of the
body to 0.8L at the tail. The stride length
s is
approximately 0.5L.
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Flow in the vicinity of the fish body
Flow generated by the body
A swimming eel generates substantial flows from the crests to the troughs
of the body wave along the full length of its body
(Fig. 3). In the medio-frontal
plane of the body, these flows form semicircles that travel posteriorly with
the body wave. In a fluid-based reference system, the flows from crests to
posterior troughs are strong, unlike the flows from crests to anterior
troughs. As the body wave amplitude increases posteriorly, so do the flows.
The peak flow speeds in the crests increase almost linearly from values close
to 0 directly behind the head to 0.014 m s-1 (0.12U) at
the tail (Fig. 4). In the
troughs, the flow speeds increase to values of up to 0.023 m s-1
(0.19U) at the tail. Contralateral semicircular flows have the same
sense of rotation. Combined, they resemble the potential region of a vortex
that has its centre within the eel's body, their flow driven by the pressure
difference between crests and troughs. With each high-pressure flow off a
crest being followed by a low-pressure flow into the neighbouring trough, the
maximum size of this vortical structure is limited to a quarter of the body
wave length. The `vortex' centre is located between the crest and the trough
of the body wave in the regions of elevated vorticity adjacent to the eel's
body. This vortical flow travels down the body along with the body wave and is
ultimately shed in the wake as a vortex. While still travelling along the
body, we call this structure a `protovortex'. Once it has been shed, it is
called a `body vortex'. The term `bound vortex' is avoided because our
experimental evidence does not show conclusively that the observed phenomenon
is a vortex or that it is in any way equivalent to a `bound vortex', which is
a free vortex core `buried' inside a lift-generating aerofoil to satisfy the
KuttaJoukowsky theorem.
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Blickhan et al. (Blickhan et al.,
1992) and Videler et al.
(Videler et al., 1999
) suggest
that the undulatory pump mechanism is driven by a combination of a travelling
wave with a posteriorly increasing amplitude envelope. The centre of the
`proto-vortex' is therefore correlated with the transition point between
concave and convex body curvature. Triantafyllou et al. (Triantafyllou et al.,
2000
) find in their
computational flow fields that local peaks in the boundary layer vorticity
occur near the nodes of the body wave. To test these proposed links between
body kinematics and flow, the position of the vortex centre was tested against
the position of the following three body wave parameters: (i) the node, where
the body midline crosses the mean path of motion; (ii) the maximum, where the
body wave reaches local extremes of lateral excursion; and (iii) the
inflection points, where the direction of the body curvature changes. The
centre of the `proto-vortex' was assumed to be at the local vorticity peak. We
found that, within the limited spatial resolution of our flow fields (3 mm),
none of the above body wave parameters correlates tightly with the position of
the vortex centre. Instead, the vortex centre shifts from a position close to
the lateral maximum when near the head until it almost coincides with the
position of the node and inflection point when it is shed off the tail
(Fig. 5, Fig. 6). The phase between node
and vortex centre induces a lateral offset that decreases with decreasing
phase. Ultimately, the `proto-vortex' crosses the mean path of motion together
with the inflection point and is shed off the tail on the opposite side
(Fig. 6, t=0.48 s;
t=0 s is defined in Fig.
3) from where the vortex was first observed at approximately
0.3L behind the head (Fig.
3, t=0.04 s).
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To test whether the observed vorticity distribution is consistent with a
simple lift generation model (Lighthill's slender body model: Lighthill,
1960), we also calculated the
lateral velocity V(t,x) of the body relative to the fluid. A
body of constant cross-sectional area A swimming in water of density
creates instantaneous lift forces FL at each body
segment according to:
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At the centre of the `proto-vortex', V(t,x) should be zero. The precision of zero positions of V(t,x) decreases anteriorly from less than 1 mm at the tail to 10 mm near the head. Within these limits, the points along the midline where V(t,x) is zero coincide with the positions of local vorticity maxima only for the counterclockwise vortices. For the clockwise vortices, there is a variable phase shift between the two. These tentative results, together with predictions of an analytical body wave model (see Appendix), suggest that body wave parameters are not suitable for predicting the body flows directly. In the present study, a simple two-dimensional pressure-based model yields more reliable predictions, but its power is limited because it ignores three-dimensional effects and wake interaction.
Flow generated at the tail
The flow at the tail results from the interaction between the body flow
reaching the tail and the flow generated by the oscillating movement of the
tail. Over most of the tail-beat cycle, it has a strong lateral component.
This is mainly a result of the body-generated low-pressure flows periodically
entering the tail region. Averaged over one tail excursion from right to left,
flow velocities are 0.019±0.004 m s-1
(0.16±0.03U) (N=23 velocity vectors from
t=0.08 s in Fig. 3 to
t=0.24 s in Fig. 6).
Flow velocities in the grid cells adjacent to the tail vary over half a
tail-beat cycle from 0.014±0.006 m s-1
(0.12±0.05U) (N=27 velocity vectors from three
tail-beats from t=0 s in Fig.
3 to t=0.64 s in Fig.
6) when the tail crosses the mean path of motion to
0.021±0.008 m s-1 (0.18±0.07U)
(N=14 velocity vectors from three tail-beats) near the stroke
reversal. The peak velocities coincide with the body flows entering the tail
region. At this moment, the mean angle of attack of the tail (the angle
between the flow and the tail fin) is approximately 50° (49±5°,
N=3 tail-beats), and a vortex is shed off the tail
(Fig. 3, t=0.12 s;
Fig. 6, t=0.32 s).
Body vortices shed to the right of the animal just before stroke reversal
rotate clockwise; body vortices shed to the left rotate counterclockwise.
Each time the tail reaches a lateral extreme and changes direction, a startstop vortex is shed at the tip of the tail. This vortex is visible in the wake as a circulating flow around an area of elevated vorticity (e.g. Fig. 6, t=0.24 s and 0.44 s). Tail vortices shed to the right of the animal rotate counterclockwise (Fig. 6, t=0.44 s); vortices shed to the left of the animal rotate in a clockwise direction (Fig. 6, t=0.24 s). In the following, startstop vortices shed at stroke-reversal from the trailing edge of the tail will be called tail vortices. Because of the limited spatial resolution of our flow fields, a body vortex is not always distinguishable from the previously shed tail vortex during and immediately after shedding (Fig. 6, t=0.24-0.36 s). However, both emerge as separate and individually recognisable body and tail vortices in the wake by the time the next body vortex is shed (Fig. 6, t=0.40 s).
Flow in the wake
Wake morphology and wake generation
In all sequences with a clearly visible wake, the flow field in the
medio-frontal plane behind a swimming eel consists of four vortices per
tail-beat cycle (Fig. 5,
Fig. 6). Consecutively shed
ipsilateral vortices have an opposite sense of rotation. This induces a strong
jet flow between them. The jet has only a small component in the direction
opposite to the swimming direction, but is directed away from the mean path of
motion at almost a right angle (=70±10°, N=3
tail-beats from t=0 s in Fig.
3 to t=0.64 s in Fig.
6). This suggests that the eel is generating considerable side
forces and little thrust. The flow speeds of the jet are 0.016±0.005 m
s-1 (0.13±0.05U; N=3 tail-beats). The
vortices move away from the mean path of motion, and the distance between the
contralateral vortices roughly doubles within 0.5 s.
A pattern of double vortices in a double row could be a cross section through either of the proposed wakes (Fig. 1). First, if the wake behind the fish is a single chain of vortex rings (Fig. 1A, Fig. 7A), a frontal cross section of such a vortex chain off the medial plane would consist of two rows of vortex pairs. Two consecutive ipsilateral vortices would have the same rotational sense. The jet through the centre of the vortex rings would appear to meander between consecutive counterrotating vortices on opposite sides of the mean path of motion. Second, two ipsilateral counter-rotating vortices could be a mediofrontal cross section through a vortex ring with a jet through the centre of the ring. The three-dimensional impression of such a wake would be a double row of vortex rings (Fig. 1B, Fig. 7B). These individual vortex rings would move away from the mean path of motion at their self-induced velocity.
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In our flow fields, consecutive ipsilateral vortices are counter-rotating. The distance between contralateral vortices perpendicular to the mean path of motion increases over time, while the distance along the mean path of motion remains constant. On the basis of these observations, we assume that consecutive ipsilateral vortices represent cross sections through a vortex ring with a diameter that can be approximated by the distance between the two vortices (Fig. 7B).
One vortex ring comprises consecutively shed ipsilateral body and tail
vortices. A clockwise tail vortex combines with the previously shed
counterclockwise body vortex to form a vortex ring to the left of the mean
path of motion, and vice versa for vortex rings to the right of the
mean path of motion. Body and tail vortices can be distinguished by their
position relative to the path of the tail tip
(Fig. 2A). Tail vortices are
shed just after the point of maximum lateral displacement of the tail, and
body vortices are shed before this point close to the mean path of motion. The
positions of the vortices reflect the swimming kinematics. The distance
between the centres of consecutive tail vortices and between consecutive body
vortices along the mean path of motion is 22±4 mm (N=3
tail-beats from t=0 s in Fig.
3 to t=0.64 s in Fig.
6), which matches half the stride length s
(
s=46±5 mm, N=3 tail-beats). Vortices of
the same type do not move apart along the mean path of motion during the
observation period of 1 s. The initial distance between shedding positions of
consecutive tail vortices perpendicular to the path of motion is
20.1±4.3 mm (N=3 tail-beats). This corresponds to twice the
tail-beat amplitude A (A=9.3±1.0 mm, N=3
tail-beats). The distance between consecutive contralateral tail vortices
almost doubles during the first second after shedding. The body vortices move
apart perpendicular to the mean path of motion at the same rate. However, at
2.6±2.1 mm (N=3 tail-beats), the initial distance between
consecutive contralateral body vortices is considerably smaller than twice the
tail-beat amplitude and is closer to the distance between consecutive
inflection points as they reach the tail tip (4.4±3.2 mm, N=3
tail-beats).
If the circulation and radius R of a circular vortex ring
are known, the vortex ring impulse I can be computed according to
(Spedding et al., 1984
):
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Immediately after shedding, body and tail vortices have a similar strength, which varies between rings from 0.0003 to 0.0007 m2s-1. This circulation is equivalent to a vortex force of 0.4-1 mN. At a momentum angle of 70°, this corresponds to a net thrust of 0.1-0.3 mN.
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Discussion |
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At slips smaller than unity, the swimmer transfers momentum when the lateral movements of its body accelerate the adjacent water. This generates longitudinal flows that are driven by the pressure gradient between ipsilateral troughs and crests of the body wave. With the body wave amplitude increasing towards the tail, this momentum transfer takes place along the entire body; as the body wave amplitude increases, so do the longitudinal flows. The localised pressure gradients not only cause flows from body wave crests to posterior troughs, but also cause a backflow from high-pressure to anterior low-pressure zones. This backflow, in turn, causes a thickening of the boundary layer in a region along the body approximately at the core of the `proto-vortex'. This thickening effect becomes visible only in slender swimmers, for which the flows are stronger than in fusiform swimmers; in these areas of backflow, the boundary layer occupies a large enough proportion of a grid cell to elevate its average vorticity significantly.
Computational studies on fusiform swimmers (Triantafyllou et al.,
2000) predict strong
transverse flows out of the mediofrontal plane near the nodes of the body wave
and predominantly longitudinal flows where the body wave reaches a lateral
maximum. From our two-dimensional flow fields, we can calculate divergence
(Stamhuis and Videler, 1995
)
to estimate the absence or presence of out-of-plane flows in incompressible
fluids. We observe weak divergence at the head and slightly higher values in
the tail region. The caudal out-of-plane flows concentrate in the areas of
high flow velocity between the node and the lateral extreme of the body
wave.
Eels generate more pronounced longitudinal flows than the previously
studied fusiform swimmers for several reasons. First, they have a larger body
wave amplitude and a shorter body wave length. Both aspects can contribute to
higher lateral body velocities. Second, bodies of almost uniform width
displace water in a different way from strongly tapered fusiform bodies. In a
fusiform body, the lateral excursion of the body at the crest of a body wave
does not significantly exceed the initial displacement of water at the head.
Noticeable crests develop only in the peduncle region
(Fig. 8A). Hence, for fusiform
undulating bodies, the water displacement and the momentum transfer are
largest at the troughs and only weak at the crests
(Müller et al.,
1997; Wolfgang et al.,
1999
). Slender fish such as
eels generate substantial lateral movement beyond the initial head
displacement along the entire body (Fig.
8B). Hence, significant pressure gradients build up in the crests
as well as the troughs of the body wave.
|
These flows along the body contain the momentum transferred to the water by the body, but they cannot be equated with the body's contribution to thrust production and simply be added to the tail's thrust output: the body flows interact with the tail flows and modify the tail's performance.
Wake generation in undulatory swimmers
As the body flows reach the tail, they cause the shedding of a body vortex
off the trailing edge of the tail tip. Independently from the shedding of this
body-generated vortex, startstop vortices are shed off the tail every
time the tail reaches a lateral extreme and changes direction. The phase
between body and tail movements determines whether consecutive body and tail
vortices are shed separately, as in our eel, or merge into one vortex, as in
our previous experiments with mullet (Müller et
al., 1997). This sensitivity
of the wake shape to the phase relationship between two vortex generators was
established by Gopalkrishnan et al. (Gopalkrishnan et al.,
1994
) and Streitlien et al.
(Streitlien et al., 1996
),
including its relevance for maximising thrust or efficiency.
In the case of our eel, the `proto-vortex' travels down the body. It
crosses the mean path of motion before it is shed as a body vortex off the
tail tip on the contralateral side of where it was first observed. We observe
the following shedding pattern in the mediofrontal cross section of the wake.
The body sheds a single row of alternating vortices close to the mean path of
motion. These lie inside a reverse von
Kármán vortex
street shed by the tail. Consecutive body and tail vortices counter-rotate and
form one vortex pair per half tail-beat cycle. The circulations of the body
and the tail vortex are similar immediately after shedding, indicating that
body vortices contribute equally to the wake and probably to thrust. In
previous studies on mullet (Müller et al.,
1997) and danios (Wolfgang et
al., 1999
), the body vortex is
shed together with a tail vortex of the same rotational sense. Body and tail
vortices combine into a single vortex. The resulting cross section through the
wake consists of a reverse von
Kármán vortex
street with two single rows of counter-rotating vortices.
Wake morphology is the result of the interaction between the circulation
building up around the tail and the momentum generated by the undulatory pump
action of the body. This interaction depends on the phase relationship between
the tail-beat cycle and the body wave travelling down the body. It can be
shown analytically (see Appendix) that a non-uniform amplitude envelope causes
a phase shift between the lateral displacement function of the body and the
pressure difference across the midplane of the fish. Assuming a constant body
wave length, the phase shift causes the `proto-vortex' to be shed off the tail
not on the mean path of motion, but offset laterally. A phase shift can be the
result either solely of a non-uniform amplitude envelope or of the interaction
between the former and a changing body wave length along the body. The
sensitivity of the phase relationship to slight changes in the amplitude
envelope and the wave length might explain how undulatory swimmers can produce
wakes with single vortices and vortex pairs at similar U/V ratios.
Eel-type swimmers can adapt their kinematics to produce a double (Hertel,
1966; Rayner,
1995
; present study) or single
(Gray, 1968
) vortex wake.
Saithe-type swimmers have been reported to produce vortex-pair wakes (Rosen,
1959
) as well as single-vortex
wakes (Aleyev, 1977
; Blickhan
et al., 1992
;
Müller et al.,
1997
; Wolfgang et al.,
1999
).
Some data exist in the literature mapping the phase between lateral
displacement function and body curvature, and several studies did indeed find
a variety of phase relationships in several undulatory swimmers (Gillis,
1997; Jayne and Lauder,
1995
; Katz and Shadwick,
1998
;
Müller et al.,
1997
; Videler and Hess,
1984
). These experimental data
on the variable phase within body wave parameters such as node and
inflection point make it likely that a similar variability exists
between these body wave parameters and the body-generated flow patterns.
However, the experimental evidence for a correlation between body wave
parameters and body vortex shedding is far from conclusive. Data on the
relationship between vortex shedding and body kinematics are still very sparse
(Müller et al.,
1997
; Wolfgang et al.,
1999
).
Experimental and theoretical approaches alike suggest that fish might change their body wave to cause changes in the phase relationship between body and tail movements, which in turn affect the phase between body and tail flow. The result is a fine control over the interaction between body and tail vortices and the vortex shedding pattern that reflects the optimisation criteria employed by the fish at each moment. The range of flow patterns observed behind undulatory swimmers to date can be explained in terms of the proposed control over body and tail vorticity. The wake of an undulatory swimmer will be a strong reverse von Kármán vortex street only if the fish is maximising thrust. When maximising efficiency, the resulting wake is a weak reverse von Kármán vortex street, which can verge on a drag wake depending on the relative strength of the merging counter-rotating body and tail vortices. Maximising thrust or efficiency requires a specific phase relationship between body and tail vortex shedding. For a phase relationship midway between the two optima, yet another wake pattern is generated. This is the vortex-pair wake observed in the present study.
Concluding remarks
The morphology of the wake and the flow adjacent to the eel's body support
the hypothesis that the eel employs its whole body to generate thrust. The
maximum flow velocities adjacent to the body increase approximately linearly
from head to tail. The tail-beat kinematics is only one factor besides the
body wave that determines the shape of the wake. The shape of the wake
deviates strongly from an optimised wake for maximising hydrodynamic
efficiency or thrust, as suggested by Triantafyllou et al. (Triantafyllou et
al., 1991; Triantafyllou et
al., 1993
) and observed in
mullet (Müller et al.,
1997
) and trout (Blickhan et
al., 1992
). Instead, it
resembles the flow pattern described by Gopalkrishnan et al. (Gopalkrishnan et
al. 1994
) for a phase
relationship between maximum thrust and efficiency, where neither is
particularly high.
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Appendix |
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Let the midline displacement, relative to the mean path of motion
x, be h(t,x) and the lateral velocity of the
midline V(t,x) be:
Uniform amplitude envelope
h=Acosk(x-Vt), where V is the wave
speed and k is the wave number. Then
Linearly increasing amplitude envelope
h=A(x/L)cosk(x-Vt), where L is
the body length of the fish. Then:
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Acknowledgments |
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References |
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