On the Stability of a System of Two Identical Point Vortices and a Cylinder

2020 ◽  
Vol 310 (1) ◽  
pp. 25-31
Author(s):  
A. V. Borisov ◽  
L. G. Kurakin
Keyword(s):  
Point Vortices ◽  
Identical Point ◽  
The Stability

scholarly journals Evolution of localized vortices in the presence of stochastic perturbations

2006 ◽  
Vol 13 (1) ◽  
pp. 41-51 ◽  
Author(s):  
E. Bécu ◽  
V. Pavlov
Keyword(s):  
Initial Distribution ◽  
Point Vortices ◽  
Symmetric Distribution ◽  
Axially Symmetric ◽  
Stochastic Perturbations ◽  
Integral Invariants ◽  
Identical Point ◽  
Runge Kutta

Abstract. We consider the evolution of a distribution of N identical point vortices when stochastic perturbations in the Hamiltonian are present. It is shown that different initial configurations of vorticity with identical integral invariants may exist. Using the Runge-Kutta scheme of order 4, it is also demonstrated that different initial configurations with the same invariants may evolve without having any tendency to approach to a unique final, axially symmetric, distribution. In the presence of stochastic perturbations, if the initial distribution of vortices is not axially symmetric, vortices can be trapped in certain domains whose location is correlated with the configuration of the initial vortex distribution.

scholarly journals On the equilibrium and stability of a row of point vortices

1995 ◽  
Vol 290 ◽  
pp. 167-181 ◽  
Author(s):  
Hassan Aref
Keyword(s):  
Vortex Dynamics ◽  
Stability Problem ◽  
Regular Polygon ◽  
Symmetric Matrix ◽  
Classical Problem ◽  
Point Vortices ◽  
Instability Modes ◽  
Systematic Enumeration ◽  
The Stability ◽  
Accessible Form

The equilibrium and stability of a single row of equidistantly spaced identical point vortices is a classical problem in vortex dynamics, which has been addressed by several investigators in different ways for at least a century. Aspects of the history and the essence of these treatments are traced, stating some in more accessible form, and pointing out interesting and apparently new connections between them. For example, it is shown that the stability problem for vortices in an infinite row and the stability problem for vortices arranged in a regular polygon are solved by the same eigenvalue problem for a certain symmetric matrix. This result also provides a more systematic enumeration of the basic instability modes. The less familiar theory of equilibria of a finite number of vortices situated on a line is also recalled.

scholarly journals On the action of viscosity in increasing the spacing ratio of a vortex street

1936 ◽  
Vol 154 (881) ◽  
pp. 67-89 ◽  
Keyword(s):  
Viscous Fluid ◽  
Point Vortices ◽  
Vortex Street ◽  
Von Karman ◽  
Spacing Ratio ◽  
The Subject ◽  
The Stability ◽  
Von Kármán ◽  
The Ideal ◽  
Stable Vortex

1—von Kármán, by considering two parallel rows (of indefinite extent) of isolated, equal, point-vortices existing in a non-viscous fluid, has shown that the only stable vortex arrangement is the asymmetrical staggered one; and then only provided that the geometry of the system is such that h / a = 0·281, where h = width between the rows, and a = distance between consecutive vortices in one row. Since von Kármán’s investigation was published, writers on the subject have attempted to connect up the street with an obstacle producing it; and to investigate the effect of channel walls upon the stability and spacing ratio of the ideal street. At the same time efforts have been made to verify von Kármán’s spacing prediction by experiment, and to check the theoretical conclusions concerning the effect of parallel walls ;§ but the results have been far from satisfactory.

Bilinear Relative Equilibria of Identical Point Vortices

2012 ◽  
Vol 22 (5) ◽  
pp. 849-885 ◽  
Author(s):  
H. Aref ◽  
P. Beelen ◽  
M. Brøns
Keyword(s):  
Relative Equilibria ◽  
Point Vortices ◽  
Identical Point

scholarly journals Close pairs of relative equilibria for identical point vortices

2011 ◽  
Vol 23 (5) ◽  
pp. 051706 ◽  
Author(s):  
Tobias Dirksen ◽  
Hassan Aref
Keyword(s):  
Relative Equilibria ◽  
Point Vortices ◽  
Identical Point ◽  
Close Pairs

Motion of Two Identical Point Vortices in a Simple Shear Flow

1985 ◽  
Vol 54 (11) ◽  
pp. 4069-4072 ◽  
Author(s):  
Yoshifumi Kimura ◽  
Hidenori Hasimoto
Keyword(s):  
Shear Flow ◽  
Simple Shear ◽  
Point Vortices ◽  
Simple Shear Flow ◽  
Identical Point

scholarly journals The finite-dipole dynamical system

2012 ◽  
Vol 468 (2146) ◽  
pp. 3006-3026 ◽  
Author(s):  
Andrew A. Tchieu ◽  
Eva Kanso ◽  
Paul K. Newton
Keyword(s):  
Equations Of Motion ◽  
Point Vortices ◽  
Inviscid Fluid ◽  
Straight Line ◽  
Single Dipole ◽  
Gradient Based ◽  
Finite Distance ◽  
The Difference ◽  
The Stability ◽  
Induced Velocity

The notion of a finite dipole is introduced as a pair of equal and opposite strength point vortices (i.e. a vortex dipole) separated by a finite distance. Equations of motion for N finite dipoles interacting in an unbounded inviscid fluid are derived from the modified interaction of 2 N independent vortices subject to the constraint that the inter-vortex spacing of each constrained dipole, ℓ, remains constant. In the absence of all other dipoles and background flow, a single dipole moves in a straight line along the perpendicular bisector of the line segment joining the two point vortices comprising the dipole, with a self-induced velocity inversely proportional to ℓ. When more than one dipole is present, the velocity of the dipole centre is the sum of the self-induced velocity and the average of the induced velocities on each vortex comprising the pair due to all the other dipoles. Each dipole orients in the direction of shear gradient based on the difference in velocities on each of the two vortices in the pair. Several numerical experiments are shown to illustrate the interactions between two and three dipoles in abreast and tandem configurations. We also show that equilibria (multi-poles) can form as a result of the interactions, and we study the stability of polygonal equilibria, showing that the N =3 case is linearly stable, whereas the N >3 case is linearly unstable.

scholarly journals Stationary states of identical point vortices and vortex foam on the sphere

2013 ◽  
Vol 469 (2150) ◽  
pp. 20120622 ◽  
Author(s):  
Kevin A. O'Neil
Keyword(s):  
Initial Conditions ◽  
Point Vortex ◽  
Intermediate Energy ◽  
Point Vortices ◽  
Stationary States ◽  
Vortex Sheets ◽  
Identical Point ◽  
Equilibrium Configurations ◽  
Stationary Vortex ◽  
Point Vortex Equilibria

Stationary configurations of identical point vortices on the sphere are investigated using a simple numerical scheme. Configurations in which the vortices are arrayed along curves on the sphere are exhibited, which approximate equilibrium configurations of vortex sheets on the sphere. Other configurations (found after starting from random initial conditions) exhibit net-like distributions of vorticity, dividing the sphere into many cells that contain no vorticity or diffuse vorticity and forming a stationary ‘vortex foam’ on the sphere. They may be viewed as intermediate-energy elements in the set of all identical point vortex equilibria on the sphere. In the continuum limit, these foam states may correspond to stationary states of multiple intersecting vortex sheets. Stationary configurations of point vortices are not found to have this character when vortices of opposite circulations are included.

scholarly journals On bifurcation of the four Liouville tori in one generalized integrable model of the vortex dynamics

2019 ◽  
Vol 487 (4) ◽  
pp. 376-380
Author(s):  
P. E. Ryabov
Keyword(s):  
Bifurcation Diagram ◽  
Vortex Dynamics ◽  
Integrable Model ◽  
Vortex Pair ◽  
Einstein Condensate ◽  
Point Vortices ◽  
Physical Parameters ◽  
Bose Einstein Condensate ◽  
Liouville Tori ◽  
The Stability

The article deals with a generalized mathematical model of the dynamics of two point vortices in the Bose-Einstein condensate enclosed in a harmonic trap, and of the dynamics of two point vortices in an ideal fluid bounded by a circular region. In the case of a positive vortex pair, which is of interest for physical experimental applications, a new bifurcation diagram is obtained, for which the bifurcation of four tori into one is indicated. The presence of bifurcations of three and four tori in the integrable model of vortex dynamics with positive intensities indicates a complex transition and the connection of bifurcation diagrams of both limit cases. Analytical results of this publication (the bifurcation diagram, the reduction to a system with one degree of freedom, the stability analysis) form the basis of computer simulation of absolute dynamics of vortices in a fixed coordinate system in the case of arbitrary values of the physical parameters of the model (the intensities, the vortex interaction and etc.).

Sign in / Sign up

Export Citation Format

Share Document