Relative equilibria of point vortices and linear vortex sheets

2018 ◽  
Vol 30 (10) ◽  
pp. 107101 ◽  
Author(s):  
Kevin A. O’Neil
Keyword(s):  
Relative Equilibria ◽  
Point Vortices ◽  
Vortex Sheets

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.

scholarly journals Point vortices on the sphere: Stability of symmetric relative equilibria

2011 ◽  
Vol 3 (4) ◽  
pp. 439-486 ◽  
Author(s):  
Frederic Laurent-Polz ◽  
◽  
James Montaldi ◽  
Mark Roberts ◽  
◽  
...  
Keyword(s):  
Relative Equilibria ◽  
Point Vortices

A versatile taxonomy of low-dimensional vortex models for unsteady aerodynamics

2018 ◽  
Vol 858 ◽  
pp. 917-948 ◽  
Author(s):  
Darwin Darakananda ◽  
Jeff D. Eldredge
Keyword(s):  
Large Scale ◽  
Bluff Body ◽  
Point Vortex ◽  
Unsteady Aerodynamics ◽  
Shear Layers ◽  
Point Vortices ◽  
The Body ◽  
Vortex Sheets ◽  
Proposed Model ◽  
Low Dimensional

Inviscid vortex models have been demonstrated to capture the essential physics of massively separated flows past aerodynamic surfaces, but they become computationally expensive as coherent vortex structures are formed and the wake is developed. In this work, we present a two-dimensional vortex model in which vortex sheets represent shear layers that separate from sharp edges of the body and point vortices represent the rolled-up cores of these shear layers and the other coherent vortices in the wake. We develop a circulation transfer procedure that enables each vortex sheet to feed its circulation into a point vortex instead of rolling up. This procedure reduces the number of computational elements required to capture the dynamics of vortex formation while eliminating the spurious force that manifests when transferring circulation between vortex elements. By tuning the rate at which the vortex sheets are siphoned into the point vortices, we can adjust the balance between the model’s dimensionality and dynamical richness, enabling it to span the entire taxonomy of inviscid vortex models. This hybrid model can capture the development and subsequent shedding of the starting vortices with insignificant wall-clock time and remain sufficiently low-dimensional to simulate long-time-horizon events such as periodic bluff-body shedding. We demonstrate the viability of the method by modelling the impulsive translation of a wing at various fixed angles of attack, pitch-up manoeuvres that linearly increase the angle of attack from $0^{\circ }$ to $90^{\circ }$, and oscillatory pitching and heaving. We show that the proposed model correctly predicts the dynamics of large-scale vortical structures in the flow by comparing the distributions of vorticity and force responses from results of the proposed model with a model using only vortex sheets and, in some cases, high-fidelity viscous simulation.

Relative equilibria of point vortices on the hyperbolic sphere

2013 ◽  
Vol 54 (6) ◽  
pp. 063101 ◽  
Author(s):  
Seungsu Hwang ◽  
Sun-Chul Kim
Keyword(s):  
Relative Equilibria ◽  
Point Vortices

Point vortices on a sphere: Stability of relative equilibria

1998 ◽  
Vol 39 (11) ◽  
pp. 5894-5907 ◽  
Author(s):  
Sergey Pekarsky ◽  
Jerrold E. Marsden
Keyword(s):  
Relative Equilibria ◽  
Point Vortices

scholarly journals Recent Progress in the Relative Equilibria of Point Vortices — In Memoriam Hassan Aref

2013 ◽  
Vol 7 ◽  
pp. 3-12 ◽  
Author(s):  
Peter Beelen ◽  
Morten Brøns ◽  
Vikas S. Krishnamurthy ◽  
Mark A. Stremler
Keyword(s):  
Recent Progress ◽  
Relative Equilibria ◽  
Point Vortices ◽  
In Memoriam

Relative equilibria of vortex sheets

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