HOW THE BODY CONTRIBUTES TO THE WAKE IN UNDULATORY FISH SWIMMING
:
FLOW FIELDS OF A SWIMMING EEL (ANGUILLA ANGUILLA)
ULRIKE K. MÜLLER*,
JORIS SMIT,
EIZE J. STAMHUIS and
JOHN J. VIDELER
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
)
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Fig. 1. Sketch of a cross section in the medio-frontal plane of the wake behind a
steady undulatory swimmer. (A) Double row of single vortices as observed in
eel, bream, trout and mullet, which in three dimensions constitute a chain of
vortex rings (Blickhan et al.,
1992 ). (B) Double row of double
vortices as observed in zebra danio, water snake and Kuhli leach. The circles
indicate shed vortices, with arrowheads indicating the rotational sense. The
arrows indicate the jet flows.
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Fig. 2. The wake behind a steadily swimming eel. The short black arrows indicate
the flow velocity. The meandering grey arrow indicates the path of the tail
tip and the swimming direction. All eels are swimming from right to left and
have just left the field of view. The colour tiles indicate the level of
vorticity in the flow, blue for clockwise vorticity, red for counterclockwise
vorticity. Darker shades indicate higher levels of vorticity. The field of
view is 108 mmx108 mm. (A) Eel (body length L=0.08 m) swimming
at speed U=0.12 m s-1. Sequence 1, see
Table 1. Tail and body vortices
have moved away from their initial shedding position (filled and open circles,
respectively) close to the tail path (grey line). The shed vortices are
visible in the flow field as areas of elevated vorticity. (B) Eel
(L=0.10 m) swimming at U=0.14 m s-1. Sequence 2,
see Table 1. (C) Eel
(L=0.10 m) swimming at U=0.12 m s-1. Sequence 3,
see Table 1.
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Fig. 3. Flow field adjacent to an eel (body length L=0.079 m) swimming
steadily at speed U=0.121 m s-1 from the lower right to
the upper left of the field of view (sequence 1, see
Table 1). The black arrows
indicate the flow velocity. Blue shades indicate clockwise vorticity, red
shades indicate counterclockwise vorticity. Darker shades indicate higher
levels of vorticity. The flow fields are continued in
Fig. 6. The time t is
arbitrarily set to t=0 s for the first frame shown in
Fig. 3. The field of view is
108 mmx108 mm.
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Fig. 4. Variation in the flow speeds adjacent to the eel's body over one tail-beat
cycle. The values at a particular position along the body can range from close
to zero when the body segment is close to an inflection point of the midline
to maximum values when the segment is in the low- or high-pressure zone. The
maximum flow speeds, indicative of the transferred momentum, increase almost
linearly from head to tail.
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Fig. 5. Midlines of a steadily swimming eel over approximately two tail-beat cycles
in an earth-bound frame of reference (swimming speed U=0.121 m
s-1, sequence 1, see Table
1; see also Fig.
2A, Fig. 3 and
Fig. 6 for flow fields). Also
shown are the positions of several body wave parameters and the location of
maximum vorticity in the flow field. The time interval between consecutive
midlines is 0.04 s. The vertical lines at either side of the graph indicate
the edge of the recorded images. The asterisk indicates t=0 s (cf.
Fig. 3 and
Fig. 6). The purple boxes
indicate the grid cells with a local maximum in vorticity, sized to scale.
Their arrowheads indicate the sense of rotation of the shed vortex. The area
of minimum curvature (brown bar) indicates the confidence interval of the
location of the inflection point.
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Fig. 6. Instances during the time course of wake generation behind a steadily
swimming eel (speed U=0.121 m s-1, sequence 1, see
Table 1). The black arrows
indicate the flow velocity, and the colour code indicates vorticity. Blue
shades indicate clockwise vorticity, red shades indicate counterclockwise
vorticity. Darker shades indicate higher levels of vorticity. The flow fields
are a continuation of the swimming sequence in
Fig. 3. The time t is
arbitrarily set to t=0 s for the first frame shown in
Fig. 3. The field of view is
108 mmx108 mm.
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Fig. 7. Hypothetical three-dimensional wakes of an eel. (A) In a cross-sectional
view above or below the medio-frontal plane, a single vortex chain would
appear as pairs of same-sense vortex pairs to either side of the mean path of
motion. A strong jet flow meandering around the mean path of motion would be
visible between consecutive contralateral vortices. (B) A double vortex ring
wake viewed in the mediofrontal plane would appear as pairs of
counter-rotating vortices to either side of the mean path of motion. A jet
flow would form between the vortices of a pair. Our two-dimensional flow
fields are consistent with scenario B rather than A.
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Fig. 8. Body contours over one tail-beat of (A) an eel and (B) a mullet swimming
steadily at 1.4Ls-1 (slip 0.6-0.7, where L is
body length). The time between contours is 0.04 s. The light-shaded area
indicates the initial water displacement by the head. The maximum crests
(thick black line) of the body wave extend considerably beyond this initial
displacement only in the eel (grey triangles). This means that only the eel
generates strong high-pressure flows. The troughs (thick black line) recess
significantly beyond the initial displacement of the head in both fish (grey
triangles). This causes strong low-pressure flows in both species.
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© The Company of Biologists Ltd 2001
