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Von Kármán vortex streets on the sphere
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10.1063/1.3258066
/content/aip/journal/pof2/21/11/10.1063/1.3258066
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/11/10.1063/1.3258066
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Schematic of a single VKS on the sphere with an illustration of the vorticity vector where the upper ring is placed at colatitude . is the total number of vortices, is the number of vortices per ring, hence . When pole vortices are included, we have .

Image of FIG. 2.
FIG. 2.

Singular values for a single von Kármán vortex street with parameters and . (a) Without pole vortices, the nullspace dimension is one; (b) with pole vortices, the nullspace dimension is three.

Image of FIG. 3.
FIG. 3.

(a) Angular velocity as a function of for a single VKS with . (b) Additional angular velocity due to pole vortices with . As approaches , .

Image of FIG. 4.
FIG. 4.

Streamline topologies for a single VKS. The fixed parameters are and . The different topology types are obtained by varying . In (a), we begin with the degenerate case , and in (d) we end with the degenerate case . A single topology type exists in the range , as illustrated in (b) and (c). We call these Type I topologies.

Image of FIG. 5.
FIG. 5.

Streamline topologies for a single VKS with vortices at the poles. The fixed parameters are , , and . The different topology types are attained by varying from 10 to −0.02. See Fig. 6 for a north pole view of the streamline topology bifurcations in the vicinity of . (a) Type I; (b) ; (c) Type II; (d) ; (e) Type III; (f) ; (g) Type IV; (h) ; (i) Type II.

Image of FIG. 6.
FIG. 6.

North pole view of the streamline topologies for a single VKS with vortices at the poles. The fixed parameters are , , and . Here, we illustrate the streamline topology bifurcations in the vicinity of . In the range as shown in (c), a flower-shaped contour consisting of five elliptic points and five saddle points appears about the pole. Figures 6(a)–6(e) correspond to Figs. 5(e)–5(i), respectively. (a) Type III; (b) ; (c) Type IV; (d) ; (e) Type II.

Image of FIG. 7.
FIG. 7.

Diagram of a double VKS with and without pole vortices. The configuration consists of one vortex street in the northern hemisphere, and a second in the southern hemisphere, where each street consists of two symmetrically skewed -vortex rings. One ring in each hemisphere has a latitude of from its respective hemisphere’s pole; these are referred to as the -rings. The second ring in each hemisphere has an angle of , and these are referred to as the -rings.

Image of FIG. 8.
FIG. 8.

Singular values for a double von Kármán vortex street: (a) without pole vortices and (b) with pole vortices. The fixed parameters are , , and .

Image of FIG. 9.
FIG. 9.

Curves relating the pole vortex strength vs the angle ratio . The fixed parameters are and . (a) (i.e., ). (b) (i.e., ).

Image of FIG. 10.
FIG. 10.

Angular velocity as a function of for . The fixed parameter is . In (a), we illustrate the curves when , while the curves in (b) correspond to .

Image of FIG. 11.
FIG. 11.

Streamline topologies for a double VKS with vortices at the poles. The fixed parameters are , , and . The different streamline topologies are attained by varying . We use , and is increased from 0.9 to 0.996. The bifurcation point in (f) corresponds to the point at which . See Fig. 12 for a north pole view of the streamline topology bifurcations in the vicinity of . The topology types above correspond to all those observed in the range (i.e., ). (a) Type I; (b) ; (c) Type II; (d) ; (e) Type III; (f) ; (g) Type II; (h) ; (i) Type I.

Image of FIG. 12.
FIG. 12.

North pole view of the streamline topologies for a double VKS with vortices at the poles in the vicinity of . At this point, the pole vortices switch signs. The fixed parameters are , , and . Figures 12(a)–12(c) correspond to Figs. 13(e)–13(g), respectively. (a) Type III; (b) ; (c) Type III.

Image of FIG. 13.
FIG. 13.

A continuation of Fig. 11, the figures above are streamline topologies for a double VKS with vortices at the poles. The fixed parameters are again , , and . The different streamline topologies are attained by varying . We use , and is increased from 1.02 to 2. The bifurcation point in (f) corresponds to the point at which . See Fig. 14 for a north pole view of the streamline topology bifurcations in the vicinity of . The topology types above correspond to all those observed in the range (i.e., ). (a) Type I; (b) ; (c) Type II; (d) ; (e) Type IV; (f) ; (g) Type III; (h) ; (i) Type V.

Image of FIG. 14.
FIG. 14.

North pole view of the streamline topologies for a double VKS with vortices at the poles in the vicinity of . At this point, the pole vortices switch signs. The fixed parameters are , , and . In the range as shown in (c), a flower-shaped contour consisting of five elliptic points and five saddle points appears about the poles. (a) Type II; (b) ; (c) Type IV; (d) ; (e) Type III. Figures 14(a)–14(e) correspond to Figs. 13(c)–13(g), respectively.

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/content/aip/journal/pof2/21/11/10.1063/1.3258066
2009-11-20
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Von Kármán vortex streets on the sphere
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/11/10.1063/1.3258066
10.1063/1.3258066
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