Something Old, Something New: Three Point Vortices on the Plane

Stremler, Mark A.

doi:10.1134/s1560354721050038

Cited by 3 publications

(1 citation statement)

References 17 publications

(43 reference statements)

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“…This complex representation of potential flows continues to be an integral part of the recent research in fluid mechanics, both for canonical [2][3][4][5][6] and applied [7][8][9][10][11][12] flows. In particular, the complex potential is used in both classical and modern monographs about vortex dynamics [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Stability of two-dimensional potential flows using bicomplex numbers

2022

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The use of the complex velocity potential and the complex velocity is widely disseminated in the study of two-dimensional incompressible potential flows. The advantages of working with complex analytical functions made this representation of the flow ubiquitous in the field of theoretical aerodynamics. However, this representation is not usually employed in linear stability studies, where the representation of the velocity as real vectors is preferred by most authors, in order to allow the representation of the perturbation as the complex exponential function. Some of the classical attempts to use the complex velocity potential in stability studies suffer from formal errors. In this work, we present a framework that reconciles these two complex representations using bicomplex numbers. This framework is applied to the stability of the von Kármán vortex street and a generalized formula is found. It is shown that the classical results of the symmetric and staggered von Kármán vortex streets are just particular cases of the generalized dynamical system in bicomplex formulation.

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Section: Introductionmentioning

confidence: 99%

Stability of two-dimensional potential flows using bicomplex numbers

2022

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A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering

Anurag,

Goodman,

O'Grady

2024

Physics of Fluids

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We introduce a new reduction of the motion of three point vortices in a two-dimensional ideal fluid. This proceeds in two stages: a change of variables to Jacobi coordinates and then a Nambu reduction. The new coordinates demonstrate that the dynamics evolve on a two-dimensional manifold whose topology depends on the sign of a parameter κ2 that arises in the reduction. For κ2>0, the phase space is spherical, while for κ2<0, the dynamics are confined to the upper sheet of a two-sheeted hyperboloid. We contrast this reduction with earlier reduced systems derived by Gröbli, Aref, and others in which the dynamics are determined from the pairwise distances between the vortices. The new coordinate system overcomes two related shortcomings of Gröbli's reduction that have made understanding the dynamics difficult: their lack of a standard phase plane and their singularity at all configurations in which the vortices are collinear. We apply this to two canonical problems. We first discuss the dynamics of three identical vortices and then consider the scattering of a propagating dipole by a stationary vortex. We show that the points dividing direct and exchange scattering solutions correspond to the locations of the invariant manifolds of equilibria of the reduced equations and relate changes in the scattering diagram as the circulation of one vortex is varied to bifurcations of these equilibria.

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Stability of Two-dimensional Potential Flows Using Bicomplex Numbers

Kleine,

Hanifi,

Henningson

2022

Preprint

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The use of the complex velocity potential and the complex velocity is widely disseminated in the study of two-dimensional incompressible potential flows. The advantages of working with complex analytical functions made this representation of the flow ubiquitous in the field of theoretical aerodynamics. However, this representation is not used in linear stability studies, where the representation of the velocity as real vectors is preferred by most authors, in order to allow the representation of the perturbation as the complex exponential function. Some of the classical attempts to use the complex velocity potential in stability studies suffer from formal errors. In this work, we present a framework that reconciles these two complex representations using bicomplex numbers. This framework is applied to the stability of the von Kármán vortex street and a generalized formula is found. It is shown that the classical results of the symmetric and staggered von Kármán vortex streets are just particular cases of the generalized dynamical system in bicomplex formulation.

show abstract

Something Old, Something New: Three Point Vortices on the Plane

Cited by 3 publications

References 17 publications

Stability of two-dimensional potential flows using bicomplex numbers

Stability of two-dimensional potential flows using bicomplex numbers

A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering

Stability of Two-dimensional Potential Flows Using Bicomplex Numbers

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