Translating hollow vortex pairs

Crowdy, Darren; Smith, Stefan G. Llewellyn; Freilich, Daniel

doi:10.1016/j.euromechflu.2012.09.007

Cited by 20 publications

(39 citation statements)

References 12 publications

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“…The significance of the results herein to such studies has yet to be ascertained. Even the linear stability properties of the incompressible vortex streets of Crowdy & Green (2011) have yet to be computed; indeed, it is only recently that the linear stability properties of the incompressible hollow vortex pair solutions found by Pocklington (1895) were worked out (Crowdy, Llewellyn Smith & Freilich 2012). The fact that the work here gives an analytical handle on the steady compressible solutions, at least for small Mach numbers, should be of great help in facilitating a study of their linear stability.…”

Section: Discussionmentioning

confidence: 99%

The effect of core size on the speed of compressible hollow vortex streets

Crowdy

Krishnamurthy

2017

J. Fluid Mech.

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The effect of weak compressibility on the speed of steadily translating staggered vortex streets of hollow vortices in isentropic subsonic flow is studied. A small-Machnumber perturbation expansion about the incompressible solutions for staggered streets of hollow vortices found recently by Crowdy & Green (Phys. Fluids, 2011, vol. 23, 126602) is carried out; the latter solutions provide a desingularization of the classical point vortex streets of von Kármán. The first-order compressible flow correction is calculated. We employ a novel scheme based on a complex variable formulation of the compressible flow equations (the Imai-Lamla method) combined with conformal mapping theory to track the vortex shape in this free boundary problem. The analysis to find the perturbed streamfunction and compressible vortex shapes is greatly facilitated by exploiting a calculus based on use of the Schottky-Klein prime function of a conformally equivalent parametric annulus. It is found that, for a vortex street of specified aspect ratio comprising vortices of specified circulation, the vortex core size is a key determinant of whether compressibility increases or decreases the steady propagation speed (relative to the incompressible street with the same parameters) and that both eventualities are possible. We focus attention on streets with aspect ratios around 0.28, which is close to the neutrally stable case for incompressible flow, and find that a critical vortex core size exists at which compressibility does not affect the speed of the street at first order in the (squared) Mach number. Streets comprising vortices with core size below the critical value speed up due to compressibility; larger vortices slow down.

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Section: Discussionmentioning

confidence: 99%

The effect of core size on the speed of compressible hollow vortex streets

Crowdy

Krishnamurthy

2017

J. Fluid Mech.

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“…Hollow dipoles were originally so named because the fluid inside the vortex boundary is at rest. Pocklington (1895) found propagating hollow vortex dipoles and this work was revisited by Crowdy et al (2013), Baker et al (1876) found a linear array of hollow vortices and Crowdy and Green (2011) found streets of hollow vortices. The above equilibrium configurations are families that depend on a non-dimensional parameter measuring the size of the vortices, and there is a limit that approaches the point vortex configuration.…”

Section: Introductionmentioning

confidence: 94%

Desingularized propagating vortex equilibria

Smith¹

2014

Fluid Dyn. Res.

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Abstract. The correction to the propagation velocity of point vortex equilibria caused by allowing the vortices to have finite core size is calculated. A matched asymptotic expansion in the small parameter ✏, given by the ratio of the core size to the dimension of the equilibrium configuration, is carried out. The resulting velocity correction is found to be of order ✏ 4 and arises from the interaction of second-and third-order terms in the inner expansion, which are themselves forced by the strain and strain derivatives of the outer field.

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“…Zannetti & Lasagna's solution for a hollow vortex pair in a channel arises from their analysis of Chaplygin cusps with the latter not appearing explicitly in their solutions. In the present paper, we will offer a different approach to obtaining these solutions, an approach which should be emphasised has also been very successful in producing other hollow vortex solutions; for example, in [5,7,9].…”

Section: Introductionmentioning

confidence: 98%

“…and the function theory of the Schottky-Klein prime function, to reveal the families of boundary shapes of the hollow vortices in a particular configuration. The hollow vortex model has been of considerable interest in recent years and many analytical solutions describing hollow vortices in different configurations have recently been discovered: works worthy of a mention involving the hollow vortex model include, for example, [5][6][7][8][9][10][11]. In addition, Giannakidis [12] has undertaken a numerical study into the effects of placing a Sadovskii vortex (of which the hollow vortex is a special type) in a channel containing ideal fluid.…”

Section: Introductionmentioning

confidence: 99%