^{1}and Eva Kanso

^{1,a)}

### Abstract

We consider the passive locomotion of rigid bodies in inviscid point-vortex wakes. This work is motivated by a common belief that live and inanimate objects may extract energy from unsteady flows for locomotory advantages. Studies on energy extraction from unsteady flows focus primarily on energy efficiency. Besides efficiency, a fundamental aspect of energy extraction for locomotion purposes is stability of motion. Here, we propose idealized wake models using periodically generated point vortices to emulate shedding of vortices from an un-modeled moving or stationary object. We assess the stability of these point-vortex wakes and find that they are stable for a range of periods, unlike the von Kármán street model which is mainly unstable. We then investigate the dynamics of a rigid body submerged in such wakes. In particular, we calculate periodic trajectories where the rigid body “swims” passively against the flow by extracting energy from the ambient vortices. All the periodic trajectories we find are unstable. The largest instabilities reported are for elliptic bodies where rotational effects play a role in destabilizing their motion. Within the context of this model, we conclude that passive locomotion of rigid bodies in inviscid wakes is unstable. Questions as to whether passive stability can be achieved when accounting for fluid viscosity and body elasticity remain open.

This work is supported by the National Science Foundation (NSF) CAREER Award No. CMMI 06-44925 and Grant No. CCF08-11480.

I. INTRODUCTION

II. PERIODICALLY GENERATED, POINT-VORTEX WAKES

III. THRUST AND DRAG WAKES

IV. PASSIVE LOCOMOTION IN PERIODICALLY GENERATED WAKES

A. Motion in thrust wakes

B. Motion in drag wakes

V. CONCLUSIONS

### Key Topics

- Rotating flows
- 63.0
- Vortex dynamics
- 13.0
- Vortex stability
- 12.0
- Land transportation
- 10.0
- Elasticity
- 7.0

## Figures

Wake A: (a) initial condition of the vortices, at t = 0, (b) at t = T/2, the leftmost vortex is removed and a new vortex is introduced to the right. The vortices are then relabeled such that the newly introduced vortex has index 1 (c) at t = T, the process is repeated such that the final vortex configuration is identical to that at t = 0 with a net translation of |Δ A | to the right. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in Natural Locomotion in Fluids and on Surfaces, Swimming, Flying, and Sliding, S. Childress, A. Hosoi, W. W. Schultz, and J. Wang, eds., The IMA Volumes in Mathematics and its Applications, Vol. 155 (2012). Copyright 2012 by Springer Science+Business Media B.V. Reprinted by permission of Springer Science+Business Media B.V.

Wake A: (a) initial condition of the vortices, at t = 0, (b) at t = T/2, the leftmost vortex is removed and a new vortex is introduced to the right. The vortices are then relabeled such that the newly introduced vortex has index 1 (c) at t = T, the process is repeated such that the final vortex configuration is identical to that at t = 0 with a net translation of |Δ A | to the right. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in Natural Locomotion in Fluids and on Surfaces, Swimming, Flying, and Sliding, S. Childress, A. Hosoi, W. W. Schultz, and J. Wang, eds., The IMA Volumes in Mathematics and its Applications, Vol. 155 (2012). Copyright 2012 by Springer Science+Business Media B.V. Reprinted by permission of Springer Science+Business Media B.V.

Wake B: (a) initial condition of the vortices, at t = 0, (b) at t = T/2, the rightmost vortex is removed and a new vortex is introduced to the left. (c) at t = T, the process is repeated such that the final vortex configuration is identical to that at t = 0 with a net translation of |Δ B | to the left.

Wake B: (a) initial condition of the vortices, at t = 0, (b) at t = T/2, the rightmost vortex is removed and a new vortex is introduced to the left. (c) at t = T, the process is repeated such that the final vortex configuration is identical to that at t = 0 with a net translation of |Δ B | to the left.

Plots of the absolute values of the Floquet multipliers versus T for periodic solutions A and B are shown in (a) and (b), respectively. Below the dashed lines (red) are the stable regions and above that is unstable.

Plots of the absolute values of the Floquet multipliers versus T for periodic solutions A and B are shown in (a) and (b), respectively. Below the dashed lines (red) are the stable regions and above that is unstable.

Schematics showing how the 3 point vortex model introduced in solutions A and B can be thought of as a thrust wake in (a) and a drag wake in (b). In (a) the thrust wake is generated as a fish movs with velocity and flaps its tail, while in (b) the drag wake is formed as a uniform flow with velocity passes by a bluff body. Compare the configuration of the vortices in the thrust wake in (a) with Figure 1(a) and the drag wake in (b) with Figure 2(a) .

Schematics showing how the 3 point vortex model introduced in solutions A and B can be thought of as a thrust wake in (a) and a drag wake in (b). In (a) the thrust wake is generated as a fish movs with velocity and flaps its tail, while in (b) the drag wake is formed as a uniform flow with velocity passes by a bluff body. Compare the configuration of the vortices in the thrust wake in (a) with Figure 1(a) and the drag wake in (b) with Figure 2(a) .

Free interactions of three circular cylinder with wake A: (a) R = 0.002, (b) R = 0.2, and (c) R = 0.6. The initial conditions of the vortices correspond to wake A based on Gröbli's solution with T = 0.5, while the circular cylinder is initially centered at z cyl (0) = −1.27 + i0.75 and initial velocity is v cyl (0) = 1.58 + i0.07.

Free interactions of three circular cylinder with wake A: (a) R = 0.002, (b) R = 0.2, and (c) R = 0.6. The initial conditions of the vortices correspond to wake A based on Gröbli's solution with T = 0.5, while the circular cylinder is initially centered at z cyl (0) = −1.27 + i0.75 and initial velocity is v cyl (0) = 1.58 + i0.07.

Periodic trajectory of cylinder-thrust wake system. Streamlines of the fluid motion are superimposed at three time instants (a) t = 0, (b) t = T/2, and t = T. The parameter values are T = 0.5, z T/2 = 1.56 + i0.96, z T = 2.29 and the initial conditions are z 1(0) = 1, z 2(0) = i1, z 3(0) = −1, while z cyl (0) = −1.27 + i0.75 and v cyl (0) = 1.58 + i0.07. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in Natural Locomotion in Fluids and on Surfaces, Swimming, Flying, and Sliding, S. Childress, A. Hosoi, W. W. Schultz, and J. Wang, eds., The IMA Volumes in Mathematics and its Applications, Vol. 155 (2012). Copyright 2012 by Springer Science+Business Media B.V. Reprinted by permission of Springer Science+Business Media B.V.

Periodic trajectory of cylinder-thrust wake system. Streamlines of the fluid motion are superimposed at three time instants (a) t = 0, (b) t = T/2, and t = T. The parameter values are T = 0.5, z T/2 = 1.56 + i0.96, z T = 2.29 and the initial conditions are z 1(0) = 1, z 2(0) = i1, z 3(0) = −1, while z cyl (0) = −1.27 + i0.75 and v cyl (0) = 1.58 + i0.07. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in Natural Locomotion in Fluids and on Surfaces, Swimming, Flying, and Sliding, S. Childress, A. Hosoi, W. W. Schultz, and J. Wang, eds., The IMA Volumes in Mathematics and its Applications, Vol. 155 (2012). Copyright 2012 by Springer Science+Business Media B.V. Reprinted by permission of Springer Science+Business Media B.V.

Periodic trajectory of cylinder-thrust wake system. Streamlines of the fluid motion are superimposed at three time instants (a) t = 0, (b) t = T/2, and (c) t = T. The parameter values are T = 1.5, z T/2 = 1.03 + i0.95, and z T = 1.03 + i0.058. The initial conditions are z 1(0) = 1.03 + i0.058, z 2(0) = i1, z 3(0) = −1, while z cyl (0) = −1.41 + i0.75, v cyl (0) = −1.44 + i0.64. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in Natural Locomotion in Fluids and on Surfaces, Swimming, Flying, and Sliding, S. Childress, A. Hosoi, W. W. Schultz, and J. Wang, eds., The IMA Volumes in Mathematics and its Applications, Vol. 155 (2012). Copyright 2012 by Springer Science+Business Media B.V. Reprinted by permission of Springer Science+Business Media B.V.

Periodic trajectory of cylinder-thrust wake system. Streamlines of the fluid motion are superimposed at three time instants (a) t = 0, (b) t = T/2, and (c) t = T. The parameter values are T = 1.5, z T/2 = 1.03 + i0.95, and z T = 1.03 + i0.058. The initial conditions are z 1(0) = 1.03 + i0.058, z 2(0) = i1, z 3(0) = −1, while z cyl (0) = −1.41 + i0.75, v cyl (0) = −1.44 + i0.64. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in Natural Locomotion in Fluids and on Surfaces, Swimming, Flying, and Sliding, S. Childress, A. Hosoi, W. W. Schultz, and J. Wang, eds., The IMA Volumes in Mathematics and its Applications, Vol. 155 (2012). Copyright 2012 by Springer Science+Business Media B.V. Reprinted by permission of Springer Science+Business Media B.V.

Periodic trajectories for four sets of initial conditions and parameter values. In all cases, the total integration time is t = 3T, z 1(0) = 1, z 2(0) = i1, z 3(0) = −1. In (a) T = 0.5, z cyl (0) = −1.35 + i1.08, v cyl (0) = 0.96 − i0.054, z T/2 = 1.56 + i0.99, z T = 2.28 − i0.0067. In (b) T = 0.5, z cyl (0) = −1.53 + i0.059, v cyl (0) = 3.04 + i0.84, z T/2 = 1.63 + i0.95, z T = 2.36 − i0.26. In (c) T = 1, z cyl (0) = −1.26 + i0.98, v cyl (0) = −1.22 − i0.086, z T/2 = 1.22 + i0.98, z T = 1.61 − i0.011. In (d) T = 1, z cyl (0) = −1.47 + i0.38, v cyl (0) = −1.22 − i0.086, z T/2 = 1.30 + i0.91, z T = 1.67 − i0.027. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in Natural Locomotion in Fluids and on Surfaces, Swimming, Flying, and Sliding, S. Childress, A. Hosoi, W. W. Schultz, and J. Wang, eds., The IMA Volumes in Mathematics and its Applications, Vol. 155 (2012). Copyright 2012 by Springer Science+Business Media B.V. Reprinted by permission of Springer Science+Business Media B.V.

Periodic trajectories for four sets of initial conditions and parameter values. In all cases, the total integration time is t = 3T, z 1(0) = 1, z 2(0) = i1, z 3(0) = −1. In (a) T = 0.5, z cyl (0) = −1.35 + i1.08, v cyl (0) = 0.96 − i0.054, z T/2 = 1.56 + i0.99, z T = 2.28 − i0.0067. In (b) T = 0.5, z cyl (0) = −1.53 + i0.059, v cyl (0) = 3.04 + i0.84, z T/2 = 1.63 + i0.95, z T = 2.36 − i0.26. In (c) T = 1, z cyl (0) = −1.26 + i0.98, v cyl (0) = −1.22 − i0.086, z T/2 = 1.22 + i0.98, z T = 1.61 − i0.011. In (d) T = 1, z cyl (0) = −1.47 + i0.38, v cyl (0) = −1.22 − i0.086, z T/2 = 1.30 + i0.91, z T = 1.67 − i0.027. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in Natural Locomotion in Fluids and on Surfaces, Swimming, Flying, and Sliding, S. Childress, A. Hosoi, W. W. Schultz, and J. Wang, eds., The IMA Volumes in Mathematics and its Applications, Vol. 155 (2012). Copyright 2012 by Springer Science+Business Media B.V. Reprinted by permission of Springer Science+Business Media B.V.

Periodic trajectories for (a) circular cylinder of radius R = 0.2 and (b) an elliptic cylinder of semi-major axis 0.55 and semi-minor axis 0.073, i.e., both cylinders have the same area 0.04π. Both periodic trajectories are unstable when subject to a small initial perturbation as depicted in the dashed trajectories. The trajectory in (a) corresponds to the same conditions as that in Figure 6 but here shown for a total time of 3T. The trajectory in (b) has period T = 0.2 the vortices of strength +0.2, −0.2, +0.2 are located at 0.69, 0.23 + i0.96, −0.99 − i0.028, while z cyl (0) = −0.96 + i0.55, θ(0) = −0.0859, v cyl (0) = 4.56 − i0.44, .

Periodic trajectories for (a) circular cylinder of radius R = 0.2 and (b) an elliptic cylinder of semi-major axis 0.55 and semi-minor axis 0.073, i.e., both cylinders have the same area 0.04π. Both periodic trajectories are unstable when subject to a small initial perturbation as depicted in the dashed trajectories. The trajectory in (a) corresponds to the same conditions as that in Figure 6 but here shown for a total time of 3T. The trajectory in (b) has period T = 0.2 the vortices of strength +0.2, −0.2, +0.2 are located at 0.69, 0.23 + i0.96, −0.99 − i0.028, while z cyl (0) = −0.96 + i0.55, θ(0) = −0.0859, v cyl (0) = 4.56 − i0.44, .

Periodic trajectory of cylinder-drag wake system. Streamlines of the fluid motion are superimposed at three time instants. The parameter values are T = 0.93, z T/2 = 1.39 + i1.01, z T = 1 and the initial conditions are z 1(0) = 1, z 2(0) = i1, z 3(0) = −1, while z cyl (0) = −0.5 + i0.5 and v cyl (0) = 0.56 − i2.68, .

Periodic trajectory of cylinder-drag wake system. Streamlines of the fluid motion are superimposed at three time instants. The parameter values are T = 0.93, z T/2 = 1.39 + i1.01, z T = 1 and the initial conditions are z 1(0) = 1, z 2(0) = i1, z 3(0) = −1, while z cyl (0) = −0.5 + i0.5 and v cyl (0) = 0.56 − i2.68, .

Periodic trajectory of cylinder-drag wake system. Streamlines of the fluid motion are superimposed at three time instants (a) t = 0, (b) t = T/2, and t = T. The parameter values are T = 1.5, z T/2 = 1.75 + i1.01, z T = 1 and the initial conditions are z 1(0) = 1, z 2(0) = i1, z 3(0) = −1, while z cyl (0) = −0.41 + i0.48 and v cyl (0) = 1.36 − i2.66, .

Periodic trajectory of cylinder-drag wake system. Streamlines of the fluid motion are superimposed at three time instants (a) t = 0, (b) t = T/2, and t = T. The parameter values are T = 1.5, z T/2 = 1.75 + i1.01, z T = 1 and the initial conditions are z 1(0) = 1, z 2(0) = i1, z 3(0) = −1, while z cyl (0) = −0.41 + i0.48 and v cyl (0) = 1.36 − i2.66, .

Periodic trajectory of ellipse-drag wake system. Streamlines of the fluid motion are superimposed at three time instants (a) t = 0, (b) t = T/2, and (c) t = T. The parameter values are a = 0.5, b = 0.08 (area of 0.04π), T = 0.186, z T/2 = 1.05 + i0.93, z T = 1.03 − i0.04 and the initial conditions are z 1(0) = 1.03 − i0.04, z 2(0) = −0.01 + i0.93, z 3(0) = −1 − i0.06, while z cyl (0) = −1.1 + i0.5 and v cyl (0) = −0.16 + i0.32, θ(0) = 0.0295, .

Periodic trajectory of ellipse-drag wake system. Streamlines of the fluid motion are superimposed at three time instants (a) t = 0, (b) t = T/2, and (c) t = T. The parameter values are a = 0.5, b = 0.08 (area of 0.04π), T = 0.186, z T/2 = 1.05 + i0.93, z T = 1.03 − i0.04 and the initial conditions are z 1(0) = 1.03 − i0.04, z 2(0) = −0.01 + i0.93, z 3(0) = −1 − i0.06, while z cyl (0) = −1.1 + i0.5 and v cyl (0) = −0.16 + i0.32, θ(0) = 0.0295, .

## Tables

Floquet multipliers of periodic trajectories.

Floquet multipliers of periodic trajectories.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content