Collapse and concentration of vortex sheets in two-dimensional flow

O’Neil, Kevin A.

doi:10.1007/978-90-481-8584-9_6

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…Numerically, some solutions have been computed before. O'Neil [27,28] used point vortices to approximate the vortex sheet and compute uniformly rotating solutions and Elling [12] constructed numerically self-similar vortex sheets forming cusps. O'Neil [29,30] also found numerically steady solutions which are combinations of point vortices and vortex sheets.…”

Section: Stationary and Rotating Solutionsmentioning

confidence: 99%

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

Gómez-Serrano

Park

Shi

et al. 2021

Commun. Math. Phys.

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In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $$\Omega $$ Ω , such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

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Section: Stationary and Rotating Solutionsmentioning

confidence: 99%

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

Gómez-Serrano

Park

Shi

et al. 2021

Commun. Math. Phys.

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“…Numerically, O’Neil [30,31] used point vortices to approximate the vortex sheet and compute uniformly rotating solutions and Elling [32] constructed numerically self-similar vortex sheets forming cusps. O’Neil [33,34] also found numerically steady solutions which are combinations of point vortices and vortex sheets.…”

Section: Introductionmentioning

confidence: 99%

Remarks on stationary and uniformly rotating vortex sheets: flexibility results

Gómez-Serrano

Park

Shi

et al. 2022

Phil. Trans. R. Soc. A.

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“…In [29] the numerical algorithm for the collapse of the vortex sheets accompanied by a strong vortex was given. In that paper, the shape of vortex sheet was initially presumed and then it was replaced by identical weak point vortices.…”

Section: Introductionmentioning

confidence: 99%

Collapse of n Point Vortices, Formation of the Vortex Sheets and Transport of Passive Markers

Kudela

2021

Energies

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In this paper, the motion of the n-vortex system as it collapses to a point in finite time is studied. The motion of vortices is described by the set of ordinary differential equations that we are able to solve analytically. The explicit formula for the solution demands the initial location of collapsing vortices. To find the collapsing locations of vortices, the algebraic, nonlinear system of equations was built. The solution of that algebraic system was obtained using Newton’s procedure. A good initial iterate needs to be provided to succeed in the application of Newton’s procedure. An unconstrained Leverber–Marquart optimization procedure was used to find such a good initial iterate. The numerical studies were conducted, and numerical evidence was presented that if in a collapsing system n=50 point vortices include a few vortices with much greater intensities than the others in the set, the vortices with weaker intensities organize themselves onto the vortex sheet. The collapsing locations depend on the value of the Hamiltonian. By changing the Hamiltonian values in a specific interval, the collapsing curves can be obtained. All points on the collapse curves with the same Hamiltonian value represent one collapsing system of vortices. To show the properties of vortex sheets created by vortices, the passive tracers were used. Advection of tracers by the velocity induced by vortices was calculated by solving the proper differential equations. The vortex sheets are an impermeable barrier to inward and outward fluxes of tracers. Arising vortex structures are able to transport the passive tracers. In this paper, several examples showing the diversity of collapsing structures with the vortex sheet are presented. The collapsing phenomenon of many vortices, their ability to self organize and the transportation of the passive tracers are novelties in the context of point vortex dynamics.

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Collapse and concentration of vortex sheets in two-dimensional flow

Cited by 4 publications

References 13 publications

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

Remarks on stationary and uniformly rotating vortex sheets: flexibility results

Collapse of n Point Vortices, Formation of the Vortex Sheets and Transport of Passive Markers

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