Relative equilibria of vortex sheets

2009 ◽  
Vol 238 (4) ◽  
pp. 379-383 ◽  
Author(s):  
Kevin A. O’Neil
Keyword(s):  
Relative Equilibria ◽  
Vortex Sheets

6—The Instability of Two Vortex Sheets Enclosing a Subsonic Jet due to Acoustic Radiation

1974 ◽  
Vol 72 (1) ◽  
pp. 79-91 ◽  
Author(s):  
N. Hardisty
Keyword(s):  
Acoustic Radiation ◽  
Vortex Sheets ◽  
Instability Waves ◽  
Subsonic Jet

SummaryThe propagation of sound in a subsonic jet separated by two vortex sheets from two semi-infinite still media is considered and it is found that instability waves arise at particular points on the vortex sheets and that their effect is confined to certain regions.

Do vortex sheets roll up?

1959 ◽  
Vol 8 (1) ◽  
pp. 77-90 ◽  
Author(s):  
Garrett Birkhoff ◽  
Joseph Fisher
Keyword(s):  
Vortex Sheets

Relative equilibria in the Bjerknes problem

2014 ◽  
Vol 55 (1) ◽  
pp. 35-48
Author(s):  
V. A. Vladimirov ◽  
A. B. Morgulis
Keyword(s):  
Relative Equilibria

Multisymplectic Relative Equilibria, Multiphase Wavetrains, and Coupled NLS Equations

2001 ◽  
Vol 107 (2) ◽  
pp. 137-155 ◽  
Author(s):  
Thomas J. Bridges ◽  
Fiona E. Laine‐Pearson
Keyword(s):  
Relative Equilibria

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

2021 ◽  
Author(s):  
Javier Gómez-Serrano ◽  
Jaemin Park ◽  
Jia Shi ◽  
Yao Yao
Keyword(s):  
Angular Velocity ◽  
Disjoint Union ◽  
Vortex Sheet ◽  
Vortex Sheets ◽  
Smooth Curves ◽  
Rigidity Results ◽  
Positive Vorticity ◽  
Constant Vorticity ◽  
Concentric Circles ◽  
Finite Disjoint Union

AbstractIn 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.

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

Energies ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 943
Author(s):  
Henryk Kudela
Keyword(s):  
Differential Equations ◽  
Point Vortex ◽  
Optimization Procedure ◽  
Vortex Sheet ◽  
Point Vortices ◽  
Vortex Sheets ◽  
Impermeable Barrier ◽  
Initial Location ◽  
Passive Tracers ◽  
Initial Iterate

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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