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Kinematics and hydrodynamics of an invertebrate undulatory swimmer: the damsel-fly larva

John Brackenbury^*

Department of Anatomy, University of Cambridge, Downing Street, Cambridge CB2 3DY, UK

View larger version (21K): [in a new window]   Fig. 1. (A) Paths traced out by the tip of the caudal fin during fast swimming in five damsel-fly larvae. Solid bars on the paths represent the orientation of the caudal fin. (B) Parameters used in the hydromechanical model (see text for details). , angle of incidence of the tail fin to the instantaneous line of travel of the fin tip; , angle of inclination of the tail fin to the mean swimming line; , perpendicular velocity of the tail tip; U, instantaneous forward velocity; V, absolute velocity of the fin tip; W, lateral velocity of the tail tip. (C) Flow parameters used in the calculation of thrust from the vortices produced during fast swimming (see text for details). R, ring radius; Ra, external radius of the vortex measured along the axis; Rp, external radius of the vortex measured along the ring plane; , momentum angle of the jet measured in the frontal plane relative to dead aft of the swimming line.

View larger version (26K): [in a new window]   Fig. 2. Kinematics of body and leg movements during slow swimming. (A) Four stages in the left-to-right half-stroke of the caudal fin. The times are with reference to the zero position of the leg tips and fin tip shown in B. (B) Paths traced out by the caudal fin tip and the tips of the hindlimbs (stars), middle limbs (open circles) and forelimbs (filled circles) during a single swimming cycle. The limb tip paths are zeroed to the start of the left-to-right swing of the abdomen. Limb tip positions are drawn relative to the head. (C) Periods of protraction (open bars) and retraction (filled bars) of the limbs during three consecutive half-strokes of the caudal fin. L, left; R, right; 1, front leg; 2, middle leg; 3, hind leg.

View larger version (31K): [in a new window]   Fig. 3. Structure of the caudal fin of Enallagma cyathigerum larva. (A) Dorsal view of the end of the abdomen bearing the three tracheal plates. Each plate receives two trunks from the abdominal tracheal network. The inset shows a cross section of the abdominal tip. Note that the base of the median plate has a quadrangular section, that of the two lateral plates a triangular section, with thickening along the ridges. (B) Lateral view of the left lateral tracheal plate. The arrows point to the flexion line separating the stiffened basal segment from the apical segment. The inset shows that the serrated appearance of the stiffening ribs is due to the presence of a single row of stout setae. (C) Configuration of the tracheal plates during swimming. Throughout the cycle, the height (h) of the apex of the fin varies from approximately 2 plate widths (configuration 1) to 3 plate widths (configuration 2). The dashed line in the side view of the tail plates represents the location of the vertical section on the right.

View larger version (19K): [in a new window]   Fig. 4. (A) Configuration of the caudal fin during the stroke cycle. The end of the abdomen is viewed from above, and the open and filled circles trace the paths of the base and tip of the fin respectively. Note how the fin bends specifically along the flexion line dividing the basal and apical segments of the tracheal plates. (B) Consecutive stages in the production of a ring vortex during a single right-to-left half-stroke of a larva viewed from above. These profiles are taken from Fig. 8, but also plot the path of the fin tip during the right-to-left and the following left-to-right half-strokes (160 ms position, far left). Arrows in the flow field indicate the axis of the vortex jet and the body flow contributing to the proximal vortex (20 ms position). (C) Whole-body profiles of a swimming larva at 20 ms intervals. Open and filled circles plot the path of the head and fin tip respectively. The stars plot the motion of a portion of dye streamer carried progressively to the right by the movements of the body as the undulatory wave travels caudally. By the time the larva has moved to the 80 ms position, the body-induced motion of particles in the dye stream joins the vortex jet released from the tail, which travels laterally in the direction of the arrow.

View larger version (19K): [in a new window]   Fig. 5. (A) Path traced out by the caudal fin tip (dashed line) during a cycle of slow swimming movements. The bars on the swimming path indicate the orientation of the fin. U, instantaneous forward velocity; t, time. (B) Instantaneous force produced by the body wave (stars and dashed line), pressure force (filled circles and broken line) produced by the wake and resultant thrust force (open circles and solid line) calculated for the swimming path shown in A using the hydromechanical model. m, virtual mass per unit length of the body; W, lateral velocity of the tail tip; , the angle of inclination of tail fin to mean swimming line; , the perpendicular velocity of the tail tip.

View larger version (81K): [in a new window]   Fig. 6. Trail produced by the wake of a larva swimming in ‘fast’ mode (note the trailing legs) just above, but making no physical contact with, a layer of milk on the bottom of the container. The fully formed trail in the right-hand panel (C) indicates two series of jets, flowing alternately to the left and right of the mean swimming line and directed to the rear. The arrows indicate the axes of two of the jets.

View larger version (62K): [in a new window]   Fig. 7. Progressive visualisation of a discrete ring vortex shed from the tail fin at the end of a half-stroke. (A) The leading edge of the vortex is being outlined by the white streamer; the arrow indicates the direction of propagation of the vortex. (B) The vortex has passed through the streamer and is drawing dye into the trailing edge. (C) Dye has progressed beyond the ring plane, and the locations of the opposite cores of the vortex can be seen. Below, drawings of the main features shown in the videographic sequence.

View larger version (108K): [in a new window]   Fig. 8. Four stages in the formation and shedding of a ring vortex during a single half-stroke of the tail fin. The larva is viewed from above and is swimming directly towards the bottom of the page along a path coinciding with a streamer of milk. The drawings outline the main features of this sequence. At the 0 ms stage (A), the tail fin is about to begin a right-to-left swing and its right surface is in contact with the streamer. As the fin moves to the animal’s left (B), it draws a current of water in its wake which, by the 40 ms stage (C), has rolled up into a ring vortex which is about to be shed. The vortex propagates to the side and to the rear in the direction of the arrow shown in the 200 ms drawing.

View larger version (24K): [in a new window]   Fig. 9. Starting manoeuvres in Enallagma cyathigerum larva. (A) Start of normal swimming sequence. On the right is a series of body profiles at 40 ms intervals together with a scheme of the body and tail flows which contribute to the first full-sized vortex (V2) in the swimming sequence. The path followed by the fin tip (filled circle) during the corresponding stages is shown on the left. A small vortex (V1) is usually produced during the second half-stroke, but the first full-sized vortex is produced on the third half-stroke from the beginning of the manoeuvre. Note that the larva moves off on a heading parallel with the original orientation of the body. (B,C) Successive profiles, at 40 ms intervals, during two rapid-flex manoeuvres, one resulting in a turn of 30° (B) and the other 160° (C) away from the original heading of the body. In each case, the larva was reacting to a stimulus (light touch) from the right-hand side. As a result of a rapid flex away from this side followed by a rapid straightening of the body, a thrust jet is produced on the right-hand side, the direction of which is approximately opposite to the line of travel of the body out of the turn.