1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Stability of passive locomotion in inviscid wakes
Rent:
Rent this article for
USD
10.1063/1.4789901
/content/aip/journal/pof2/25/2/10.1063/1.4789901
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/2/10.1063/1.4789901

Figures

Image of FIG. 1.
FIG. 1.

Wake A: (a) initial condition of the vortices, at = 0, (b) at = /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 = , the process is repeated such that the final vortex configuration is identical to that at = 0 with a net translation of |Δ| to the right. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in 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.

Image of FIG. 2.
FIG. 2.

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

Image of FIG. 3.
FIG. 3.

Plots of the absolute values of the Floquet multipliers versus 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.

Image of FIG. 4.
FIG. 4.

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) .

Image of FIG. 5.
FIG. 5.

Free interactions of three circular cylinder with wake A: (a)  = 0.002, (b)  = 0.2, and (c)  = 0.6. The initial conditions of the vortices correspond to wake A based on Gröbli's solution with = 0.5, while the circular cylinder is initially centered at (0) = −1.27 + 0.75 and initial velocity is (0) = 1.58 + 0.07.

Image of FIG. 6.
FIG. 6.

Periodic trajectory of cylinder-thrust wake system. Streamlines of the fluid motion are superimposed at three time instants (a) = 0, (b) = /2, and = . The parameter values are = 0.5, = 1.56 + 0.96, = 2.29 and the initial conditions are (0) = 1, (0) = 1, (0) = −1, while (0) = −1.27 + 0.75 and (0) = 1.58 + 0.07. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in , 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.

Image of FIG. 7.
FIG. 7.

Periodic trajectory of cylinder-thrust wake system. Streamlines of the fluid motion are superimposed at three time instants (a) = 0, (b) = /2, and (c) = . The parameter values are = 1.5, = 1.03 + 0.95, and = 1.03 + 0.058. The initial conditions are (0) = 1.03 + 0.058, (0) = 1, (0) = −1, while (0) = −1.41 + 0.75, (0) = −1.44 + 0.64. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in 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.

Image of FIG. 8.
FIG. 8.

Periodic trajectories for four sets of initial conditions and parameter values. In all cases, the total integration time is = 3, (0) = 1, (0) = 1, (0) = −1. In (a) = 0.5, (0) = −1.35 + 1.08, (0) = 0.96 − 0.054, = 1.56 + 0.99, = 2.28 − 0.0067. In (b) = 0.5, (0) = −1.53 + 0.059, (0) = 3.04 + 0.84, = 1.63 + 0.95, = 2.36 − 0.26. In (c) = 1, (0) = −1.26 + 0.98, (0) = −1.22 − 0.086, = 1.22 + 0.98, = 1.61 − 0.011. In (d) = 1, (0) = −1.47 + 0.38, (0) = −1.22 − 0.086, = 1.30 + 0.91, = 1.67 − 0.027. From B. G. Oskouei and E. Kanso, “Stability of passive locomotion in periodically-generated vortex wakes,” in 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.

Image of FIG. 9.
FIG. 9.

Periodic trajectories for (a) circular cylinder of radius = 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 3. The trajectory in (b) has period = 0.2 the vortices of strength +0.2, −0.2, +0.2 are located at 0.69, 0.23 + 0.96, −0.99 − 0.028, while (0) = −0.96 + 0.55, θ(0) = −0.0859, (0) = 4.56 − 0.44, .

Image of FIG. 10.
FIG. 10.

Periodic trajectory of cylinder-drag wake system. Streamlines of the fluid motion are superimposed at three time instants. The parameter values are = 0.93, = 1.39 + 1.01, = 1 and the initial conditions are (0) = 1, (0) = 1, (0) = −1, while (0) = −0.5 + 0.5 and (0) = 0.56 − 2.68, .

Image of FIG. 11.
FIG. 11.

Periodic trajectory of cylinder-drag wake system. Streamlines of the fluid motion are superimposed at three time instants (a) = 0, (b) = /2, and = . The parameter values are = 1.5, = 1.75 + 1.01, = 1 and the initial conditions are (0) = 1, (0) = 1, (0) = −1, while (0) = −0.41 + 0.48 and (0) = 1.36 − 2.66, .

Image of FIG. 12.
FIG. 12.

Periodic trajectory of ellipse-drag wake system. Streamlines of the fluid motion are superimposed at three time instants (a) = 0, (b) = /2, and (c) = . The parameter values are = 0.5, = 0.08 (area of 0.04π), = 0.186, = 1.05 + 0.93, = 1.03 − 0.04 and the initial conditions are (0) = 1.03 − 0.04, (0) = −0.01 + 0.93, (0) = −1 − 0.06, while (0) = −1.1 + 0.5 and (0) = −0.16 + 0.32, θ(0) = 0.0295, .

Tables

Generic image for table
Table I.

Floquet multipliers of periodic trajectories.

Loading

Article metrics loading...

/content/aip/journal/pof2/25/2/10.1063/1.4789901
2013-02-08
2014-04-19
Loading

Full text loading...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Stability of passive locomotion in inviscid wakes
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/2/10.1063/1.4789901
10.1063/1.4789901
SEARCH_EXPAND_ITEM