Motion of a two-dimensional monopolar vortex in a bounded rectangular domain

Geffen, van Jhgm Jos; Meleshko, V. V.; Heijst, van Gjf Gert-Jan

doi:10.1063/1.869024

Cited by 13 publications

(9 citation statements)

References 5 publications

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“…As the forcing is no longer applied, the flow gradually obtains an almost linear relationship between vorticity and stream function. Similar results were also found by Van Geffen, Meleshko, and Van Heijst, 19 who studied the evolution of a decaying monopolar vortex on a square domain. Although the Reynolds number in both the experiments and the simulations is initially high, the ultimate flow pattern closely resembles the fundamental mode of a decaying viscous flow in the limiting case Reϭ0.…”

Section: Discussionsupporting

confidence: 86%

Decaying quasi-two-dimensional viscous flow on a square domain

1998

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Section: Discussionsupporting

confidence: 86%

Decaying quasi-two-dimensional viscous flow on a square domain

1998

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“…The near-wall vortex structures in the flows around obstacles or cavities have been studied by Chernyshenko (1988), Fornberg (1993), Elcart et al (2000). The papers by Goldstik (1963), van Geffen, Meleshko & van Heijst (1996), Stremler & Aref (1999 and Elcart & Miller (2001) consider finite channels but in a different from our context. For example, Elcart & Miller (2001) use the reduction of the governing equations to Dirichlet's problem for the equation −∆Ψ = Ω(Ψ ) with a non-decreasing positive function Ω(Ψ ) that can be discontinuous; their result predicts a steady monopole vortex in a finite channel with the inflow of an irrotational fluid.…”

Section: Instantaneous Filtration and Steady Separated Flowsmentioning

confidence: 92%

Planar inviscid flows in a channel of finite length: washout, trapping and self-oscillations of vorticity

2010

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The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained.

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“…In the case of a no-slip condition ͑i.e., the velocity equals zero everywhere at the wall͒ viscously generated vorticity at the walls will in general drastically affect the vortices. Using free-slip walls of course also affects the motion of the monopoles but to a lesser extent than no-slip walls ͑see van Geffen et al, 24 where the effect of walls on the evolution of a single monopole is studied͒.…”

Section: Computations With a Finite-difference Methodsmentioning

confidence: 99%

Collapse interactions of finite-sized two-dimensional vortices

et al. 1997

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The point vortex model predicts that a certain configuration of three point vortices leads to a collapse of these vortices to one point. Numerical simulations have been performed to investigate the effect of a finite vortex size on this two-dimensional collapse interaction. The paper presents results obtained with contour dynamics simulations of patches of uniform vorticity, and results obtained with finite difference simulations of vortices with continuous properties. In addition, the effect of viscosity and the presence of impermeable domain boundaries are investigated. The results show that the motion of finite-sized vortices is quite similar to the motion of point vortices as long as the mutual distance between the vortices is larger than their size. When the vortices are closer together their shapes start to deform and the subsequent evolution is different from that of the point vortices, and an actual collapse to one vortex does not take place. © 1997 American Institute of Physics.

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Motion of a two-dimensional monopolar vortex in a bounded rectangular domain

Cited by 13 publications

References 5 publications

Decaying quasi-two-dimensional viscous flow on a square domain

Decaying quasi-two-dimensional viscous flow on a square domain

Planar inviscid flows in a channel of finite length: washout, trapping and self-oscillations of vorticity

Collapse interactions of finite-sized two-dimensional vortices

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