Rapid relaxation of an axisymmetric vortex

Bernoff, Andrew J.; Lingevitch, Joseph F.

doi:10.1063/1.868362

Cited by 76 publications

(80 citation statements)

References 9 publications

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“…The connection of the translation invariance of the Navier-Stokes equation with the decay associated to the first moment of the vorticity seems first to have been remarked upon by Bernoff and Lingevitch, [3]. The connection between symmetries of the linear and nonlinear heat equation and Burgers' equation and the decay rates of the long-time asymptotics of solutions of these equations was systematically explored in [31] and [23].…”

Section: Remark 418mentioning

confidence: 99%

“…To control the nonlinear terms in (2) or (12), we will need estimates on the velocity in terms of the vorticity. Let ω and u be related via the Biot-Savart law (3). If ω ∈ L p (R 2 ) for some p ∈ (1, 2), it follows from the classical Hardy-Littlewood-Sobolev inequality that u ∈ L q (R 2 ) 2 where 1 q = 1 p − 1 2 , and there exists C > 0 such that…”

Section: Smoothing and Compactness Propertiesmentioning

confidence: 99%

“…More precisely, if ω(x, t) is a solution of (2) such that ω(x, t) dx = 0, then the velocity field u(x, t) given by (3) satisfies |u(·, t)| 2 = ∞ for all t. Explicit examples of such infinite energy solutions are the so-called Oseen vortices:…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

Gallay

Wayne

2005

Commun. Math. Phys.

158

210

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Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called "Oseen's vortex". This implies that the Oseen vortices are dynamically stable for all values of the circulation Reynolds number, and our approach also shows that these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Finally, under slightly stronger assumptions on the vorticity distribution, we also give precise estimates on the rate of convergence toward the vortex.

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Section: Remark 418mentioning

confidence: 99%

Section: Smoothing and Compactness Propertiesmentioning

confidence: 99%

See 1 more Smart Citation

Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

Gallay

Wayne

2005

Commun. Math. Phys.

158

210

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“…Weak vorticity, much like a passive scalar, is subject to spiral wind-up and enhanced diffusion in the dominant axisymmetric flow field of a vortex (Lundgren 1982;Sutyrin 1989;Bernoff & Lingevitch 1994;Bassom & Gilbert 1998). However, vorticity is coupled into the flow field, and so can interact with the dynamics of the vortex.…”

Section: Introductionmentioning

confidence: 99%

Vortex motion in a weak background shear flow

2004

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A point vortex is introduced into a weak background vorticity gradient at finite Reynolds number. As the vortex spreads viscously so the background vorticity becomes wrapped around it, leading to enhanced diffusion of vorticity, but also giving a feedback on the vortex and causing it to move. This is investigated in the linear approximation, using a similarity solution for the advection of weak vorticity around the vortex, at finite and infinite Reynolds number. A logarithmic divergence in the far field requires the introduction of an outer length scale L and asymptotic matching. In this way results are obtained for the motion of a vortex in a weak vorticity field modulated on the large scale L and these are confirmed by means of numerical simulations.

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“…In the viscous spreading of a liquid drop on a flat surface, at large times the drop approaches an axisymmetric form with a unique spatial profile upon rescaling the radius and the height [68,60]. Other examples are the diffusion of a localized source of heat [74,78,51,52] and the viscous spreading of vorticity in a two-dimensional fluid where the solutions approach a spreading Gaussian profile [12,42].…”

Section: Introductionmentioning

confidence: 99%

Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

Self Cite

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Abstract. We discuss a methodology for studying the linear stability of self-similar solutions. We will illustrate these fundamental ideas on three prototype problems: a simple ODE with finite-time blowup, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, we will discuss the use of dimensional analysis to derive scaling invariant similarity variables, and the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions we will demonstrate that the symmetries give rise to positive eigenvalues associated with the symmetries, and show how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

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Rapid relaxation of an axisymmetric vortex

Cited by 76 publications

References 9 publications

Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

Vortex motion in a weak background shear flow

Stability and dynamics of self-similarity in evolution equations

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