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First published online August 23, 2004
Journal of Experimental Biology 207, 3265-3279 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.01139

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The hydrodynamics of eel swimming II. Effect of swimming speed

Eric D. Tytell

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

View larger version (27K): [in a new window]   Fig. 1. Swimming kinematics. Shades from white to red represent different swimming speeds, and shades of blue and green represent different individuals. In A and B, the boxes are standard statistical box plots, with the box stretching from the 25th to 75th quartile, which identifies where 50% of the data lie, and a line at the median. The error bars above and below each box reach to the maximum or minimum values or 1.5 times the size of the box, whichever is smaller. Any points that are beyond the length of the error bars are identified as outliers and shown as separate points. The narrow areas along the boxes represent approximate 95% confidence intervals. (A) Mean tail velocity, Utail, equal to 4Af, where A and f are the tail beat amplitude and frequency, respectively, against swimming speed (U). Solid line, linear regression; dotted line, slope of one. Mean Strouhal number (2Af/U) is shown for each speed. (B) Body wave speed, V, against swimming speed. Solid line, linear regression; dotted line, slope of one. Slip (U/V) is shown below each speed. (C) Undulation amplitude, defined as half the total body excursion at each point along the body at the four swimming speeds. Thickness of the line represents standard error.

View larger version (33K): [in a new window]   Fig. 2. Tail beat frequency against tail beat amplitude. Shades from white to red represent different swimming speeds, and marker shape indicates different individuals. Lines of constant 0.3 Strouhal number are shown in black. The correlation coefficient r and mean Strouhal number are shown for each swimming speed.

View larger version (34K): [in a new window]   Fig. 3. Angle of attack of the tail. Shades from white to red represent increasing swimming speeds. (A) Angle of attack plotted against tail velocity over complete tail beats. Different shaped markers represent different individuals. (B) Fraction of the tail beat cycle in which the tail has a positive angle of attack plotted against swimming speed. Shades of blue and green represent different individuals. Boxes are standard statistical box plots, described in detail in Fig. 1.

View larger version (98K): [in a new window]   Fig. 4. Wake flow at three different swimming speeds and three different phases during the tail beat cycle. Black arrows represent flow velocity magnitude and direction. Vorticity is shown in color in the background, and contours of the discriminant for complex eigenvalues at -200, -500 and -1000 are shown in white. The eel's tail is in blue at the bottom. Note that the vector scale is different for each swimming speed but the length and vorticity scales are the same in all plots. Mean jet magnitude in L s-1 is written beside each jet. Horizontal lines are provided to facilitate comparisons of jet diameter.

View larger version (19K): [in a new window]   Fig. 5. Size, strength and angle of the lateral jets in the wake at different swimming speeds. Boxes are standard statistical box plots, described in detail in Fig. 1. Shades from white to red represent different swimming speeds, and shades of blue and green represent different individuals. (A) Mean jet velocity magnitude against swimming speed. A linear regression line is shown in black and the r2 value is indicated above. (B) Mean angle of the jet against swimming speed. (C) Jet diameter against swimming speed. No significant linear relationship exists in B and C and so a regression line is not shown.

View larger version (34K): [in a new window]   Fig. 6. Relationship of the wake jet diameter to body wavelength. Shades from white to red represent different swimming speeds, and marker shape indicates different individuals. The convex hull containing each individual's points is shown with thin black lines. A linear regression line is shown with a thick black line, and a gray dotted line indicates a slope of 0.5. The P and r2 values, and the regression equation, are given in the corner.

View larger version (41K): [in a new window]   Fig. 7. Mean flow in an 8x8 mm region, 8 mm behind the tail, plotted against the mean tail speed. Filled symbols represent lateral velocities, and open symbols indicate axial velocities. Shades from white to red represent different swimming speeds, and symbol shape indicates different individuals. The linear and quadratic regression lines for axial and lateral velocities, respectively, are shown with thick black lines, and the r2 values and regression equations are indicated nearby.

View larger version (30K): [in a new window]   Fig. 8. Maximum circulation value for the primary vortex plotted against the total circulation added by the tail, estimated by equation 10. A quadratic regression line is shown with a thick black line, and a one-to-one relationship is shown with a gray broken line. Shades from white to red represent different swimming speeds, and marker shape indicates different individuals. The P and r2 values, and the regression equation, are given in the corner.

View larger version (26K): [in a new window]   Fig. 9. Log-log plot of the cost of producing the wake against mean tail speed. A linear regression is shown with a thick black line, and the P and r2 values and the regression equation are given in the corner. Shades from white to red represent different swimming speeds and marker shape indicates different individuals.

View larger version (23K): [in a new window]   Fig. 10. Comparison of impulse and power estimates at different swimming speeds. Coefficients are shown on the left-hand axes, and dimensional values are shown on the right-hand axes. Boxes are standard statistical box plots, described in detail in Fig. 1. Open boxes represent estimates from PIV, and filled boxes represent estimates from the kinematics. Colors indicate what type of estimate was used. (A) Impulse estimates. (B) Power estimates.