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First published online April 23, 2004
Journal of Experimental Biology 207, 1825-1841 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.00968

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The hydrodynamics of eel swimming : I. Wake structure

Eric D. Tytell^* and George V. Lauder

Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA

View larger version (17K): [in a new window]   Fig. 1. Flow fields behind swimming eels according to two previous studies. Red arrows indicate flow with clockwise rotation and blue arrows indicate counter-clockwise rotation. (A) Results from Müller et al. (2001), showing the wake structure they observed. Proto-vortices (dotted lines) appear to be vortices centered on the body that progress down the body. After they are shed into the wake they are shown as dashed lines. They are shed after the stop/start vortex (solid lines), resulting in two same-sign vortices being shed each tail beat. (B) Computational fluid dynamic model of Carling et al. (1998). The model indicates a large flow wrapping around the eel, resulting in a net upstream flow in the wake behind the eel.

View larger version (17K): [in a new window]   Fig. 2. Methods. Eels were filmed from below using two synchronized high-speed cameras aimed at a 45° mirror below the flow tank. A laser light sheet 7 mm above the bottom of the tank illuminated the eel's wake and part of its tail. One camera (labeled `kinematics') imaged the whole eel, and the other camera (labeled `PIV') imaged the light sheet. Representative images from each camera are shown to the left. Diagram is not to scale.

View larger version (6K): [in a new window]   Fig. 3. Coordinate system used for elongated body theory calculations. The solid line represents a midline at one time, while the dotted line represents it at a later time. Perpendicular and parallel velocity, v⊥ and v||, are shown as vectors at the point (xb, yb). The arc length s is shown along the midline and the eel is swimming into a flow U. L is total body length.

View larger version (28K): [in a new window]   Fig. 4. Flow tank boundary layer. The boundary layer thickness was 7.3 mm or less at all swimming velocities. Black boxes are standard statistical box plots, with the box stretching from the 25th to 75th quartile, a white line at the median, and whiskers of 1.5 times the interquartile range. Outliers are shown as separate points. (A) Laminar boundary layer at flow speeds less than 95 mm s-1 with fit Blasius boundary layer profile (Faber, 1995). The boundary layer thickness at 0.99U was 7.3 mm (green dotted lines). (B) Turbulent boundary layer at flow speeds above 120 mm s-1. The normalized distance y+ and velocity u+ are shown on the top and right, respectively. The law of the wall profile for turbulent boundary layers, u+=5.75 log y++5.2, is shown in red. Note that this is a semi-log plot. (C) Axial component of velocity from the horizontal light sheet, 7 mm above the bottom, showing turbulent effects. Flow is from bottom to top. Note the streamwise regions of reduced velocity. The bottom profile shows mean velocity (solid line) and mean velocity about two hours later (dotted line). A histogram of velocities (solid line) is also shown beside the color bar with a histogram from about two hours later (dotted line).

View larger version (116K): [in a new window]   Fig. 5. Representative flow field from behind an eel at 90% of the tail beat cycle. The field is a phase average of 14 tail beats. Vorticity is shown in color in the background, and contours of the discriminant for complex eigenvalues at–200,–500 and–1000 are shown in red. The eel's tail is in blue at the bottom, with red arrows, scaled in the same way as the flow vectors, which indicate the motion of the tail. Vector heads are retained on vectors shorter than 2.5 cm s-1 to show the direction of the flow.

View larger version (43K): [in a new window]   Fig. 6. Velocity transects through vortices in the eel's wake over time. The center of the first vortex is shown by the vertical dotted lines, and zero velocity is indicated by the horizontal dotted lines. Representative flow fields are shown to the right, indicating the position of the transect, with vorticity shown in color. The cross identifies the position of the first vortex, and the circle identifies the position of the second. Standard error around each velocity trace is shown by a lighter-colored region. (A) Transects through the primary vortex and, once it is formed, the secondary vortex. Idealized profiles through a single Rankine vortex and two same-sign Rankine vortices are shown in black at top and bottom. The position of the secondary vortex, plus or minus standard error, is shown as a bar along the zero line. Before the secondary vortex is completely formed, this bar indicates the position of the inflection point in velocity where the vortex will be formed. (B) Transects across the lateral jet, from the secondary vortex of one half tail beat to the primary vortex of the next. An idealized profile through a small-core vortex ring is shown in black above.

View larger version (57K): [in a new window]   Fig. 7. Flow fields close to the body of a swimming eel, shown in gray. The lateral position of the eel's snout (off the view) is shown as a black arrow. Velocities are phase averaged across 14 tail beats by interpolating the normal gridded coordinate system on to a system defined by the distance from the eel's body and the distance along the body from the head. Approximate positions of the proto-vortices, defined by Müller et al. (2001), are shown in red (clockwise rotation) and blue (counter-clockwise rotation).

View larger version (66K): [in a new window]   Fig. 8. Flow field behind the eel, averaged over 14 complete tail beats, centered on the tip of the eel's tail, shown as a black circle. Arrow heads are retained for velocities lower than 6.5 mm s-1 to indicate flow direction. Axial flow is shown in color: red is downstream and blue is upstream. Two profiles of velocity are shown in black above and below the flow field, with standard error in gray and total momentum flux represented by the trace printed beside it. Black lines across the field indicate where the velocity traces were measured (25 mm and 95 mm behind the tail). The vertical scale is the same for both traces.

View larger version (44K): [in a new window]   Fig. 9. Representative traces for force, impulse and power from large-amplitude elongated body theory (EBT; in black) and particle image velocimetry (PIV; in red and green). Each black line shows force and power for a single tail beat. A total of 14 tail beats from a single swimming bout are shown. (A) Force (left axis) and impulse (right axis) over a tail-beat cycle. Because impulse is force integrated over time, impulses are indicated as lines, showing the impulse value and the time over which it was integrated. (B) Power from EBT and PIV over a tail-beat cycle. PIV values have standard error in a lighter color around the trace. The total power measured through PIV is shown in green, and the `lateral' power, measured using only the lateral velocity component, is shown in red.

View larger version (17K): [in a new window]   Fig. 10. Schematic summary of the results of the present study, showing the wake behind a swimming eel at three different times. The size of the eel and position of the vortices are scaled to represent the true spacing. Vortices are indicated by blue and red arrows; primary vortices are solid lines and secondary vortices are dotted lines. The lateral jets are shown as block arrows, with lengths and angle proportional to the jet magnitude and direction.