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First published online September 9, 2005
Journal of Experimental Biology 208, 3519-3532 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01813

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On the estimation of swimming and flying forces from wake measurements

John O. Dabiri

Graduate Aeronautical Laboratories and Bioengineering, California Institute of Technology, Pasadena, CA 91125, USA

View larger version (12K): [in a new window]   Fig. 1. Schematic of control volume concept, indicating variables described in the text. The control volume moves along with the animal so that it is stationary in the reference frame of the animal. The boundary of the control volume encloses both the animal and the fluid with which it interacts. The vortex wake resulting from this interaction is shown trailing the animal. For definitions, see List of symbols.

View larger version (18K): [in a new window]   Fig. 2. Schematic of wake vortex parameters described in the text. The distribution of wake vorticity (x) is indicated by grey patches in this cross-section view. Hypothetical flow streamlines inside the vortex are sketched below the vortex patches. The integrals in Eq. 4 are evaluated throughout the vortex volume VV and on the vortex surface SV. The dimension S is the wake vortex width.

View larger version (142K): [in a new window]   Fig. 3. DPIV measurements of flow created by a mechanical wake generator. Images correspond to a meridian symmetry plane of the wake vortex ring. Streamlines of the velocity field (i.e. lines tangent to each vector in the velocity field) are plotted in yellow. Exit plane of the vortex generator is located at the upper margin of the frame. Flow is directed vertically downward. Flow field is 20 cm in height. (A) View of vortex ring propagating downstream from the mechanical wake generator in the laboratory reference frame. (B) Same vortex ring as in A, viewed in a reference frame that moves at the speed of the propagating vortex ring. The full extent of the vortex is clearly visible from this perspective. White arrows indicate location of vortex ring in A and B.

View larger version (16K): [in a new window]   Fig. 4. Demonstration of the method for measuring solid body added-mass (i.e. Darwin, 1953). Axisymmetric inviscid flow about a solid sphere is shown in cross section. Solutions were computed using dimensional units to allow comparison with experimental results. Sphere radius and propagation speed are 2.54 cm and 1 cm s-1, respectively. Computational domain is 50 sphere radii axially in both directions normal to the reference plane and approximately 12 cm radially. Variables d0 and rL indicate (qualitatively) finite upstream approach distance and reference plane radius, respectively. Total fluid drift (i.e. for infinite domain) was computed using measured partial drift in the computational domain and an analytical asymptotic correction factor (i.e. Eq. 12; Eames et al., 1994). (A) Sphere approaches initially planar Lagrangian surface from left; t=0 s. (B) Streamwise distortion of Lagrangian surface occurs as the sphere passes through the plane; t=4.45 s. Note that since only streamwise Lagrangian displacement is plotted, the sphere appears to pass through the plane. Plots of combined streamwise and transverse Lagrangian displacement (e.g. D-F below) verify that the plane is actually distorted around the sphere. (C) Volume between initial plane and horn-like distorted surface is the drift volume, D, from which the added-mass coefficient is computed; t=13.3 s. (D-F) Individual Lagrangian particle paths corresponding to t=0 s, 4.45 s and 13.3 s, respectively.

View larger version (19K): [in a new window]   Fig. 5. Time dependence of drift volume induced by the motion of a solid sphere through an inviscid fluid. Sphere radius and propagation speed are 2.54 cm and 1 cm s-1, respectively. Total fluid drift (i.e. for infinite domain) was computed using measured partial drift in the measurement window and an analytical asymptotic correction factor (i.e. Eq. 12; Eames et al., 1994). The oscillation in drift volume corresponds to changes in direction of Lagrangian particle translation during looping elastica trajectories.

View larger version (103K): [in a new window]   Fig. 6. Measurements of Lagrangian drift induced by translating fluid vortices. Vortex is indicated by black arrow. (A) Vortex approaches several planar Lagrangian surfaces downstream of the vortex generator (several more planes further downstream not shown); t=0.07 s. (B,C) Fluid vortices interact with Lagrangian surfaces in a manner similar to that observed for the inviscid sphere solution. Initially planar surfaces are deformed to horn-like shapes; t=3.34 s and 13.3 s, respectively. (D-F) Individual Lagrangian particle paths corresponding to t=0.07 s, 3.34 s and 13.3 s, respectively.

View larger version (30K): [in a new window]   Fig. 7. Drift and fluid body volume measurements for translating vortices. Colored solid lines: drift volume of planes initially located in 0.3 cm increments from 2.7 cm (red) to 8.0 cm (blue) downstream of the vortex generator exit. Total fluid drift (i.e. for infinite domain) was computed using measured partial drift in the measurement window and an analytical asymptotic correction factor (i.e. Eq. 12; Eames et al., 1994). Closed circles: measured volume of ellipsoidal fluid vortex. Error bars indicate measurement uncertainty. Broken blue line: least-squares linear fit to vortex volume measurements. Solid red line: linear fit required to exactly match the added-mass coefficient of an equivalent solid body. Inviscid sphere solution is plotted for comparison (black line).

View larger version (20K): [in a new window]   Fig. 8. Comparison of time-averaged and instantaneous forces and their related dynamics. Three hypothetical flying animals of equal weight (mg) generate identical time-averaged forces but different instantaneous forces and flight dynamics over three stroke cycles. Solid black line, FV-mg=0; broken blue line, FV-mg=sin(2t); dotted red line FV-mg=-sin(2t). Vertical velocity V(t) and trajectory Y(t) are determined from integration with initial conditions V(0)=0 and Y(0)=0. T, propulsive stroke duration. Figures are plotted using generic units.

View larger version (88K): [in a new window]   Fig. 9. DPIV measurements of a free-swimming Aurelia aurita jellyfish. Images are taken in the laser sheet plane, which is aligned close to the plane of symmetry of the animal. Vortex wake of the free-swimming jellyfish consists of a train of nearly axisymmetric vortex rings (white arrows; cores numbered in order of formation). Flow field is approximately 14 cm in height.