The Interaction of Two Unsteady Point Vortex Sources in a Deformation Field in 2D Incompressible Flows

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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Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos

Discrete & Continuous Dynamical Systems - B ◽

10.3934/dcdsb.2009.11.971 ◽

2009 ◽

Vol 11 (4) ◽

pp. 971-995 ◽

Cited By ~ 26

Author(s):

Xavier Perrot ◽

◽

Xavier Carton

Keyword(s):

Hamiltonian Dynamics ◽

Point Vortex ◽

Deformation Field ◽

Vortex Interaction ◽

Transition To Chaos ◽

Harmonic Resonance

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The life-cycle of tripoles in two-dimensional incompressible flows

Journal of Fluid Mechanics ◽

10.1017/s0022112094001114 ◽

1994 ◽

Vol 267 ◽

pp. 53-82 ◽

Cited By ~ 57

Author(s):

Xavier Carton ◽

Bernard Legras

Keyword(s):

Incompressible Flows ◽

Mutual Influence ◽

Bottom Topography ◽

Point Vortex ◽

Finite Amplitude ◽

Transformation Process ◽

Two Dimensional ◽

The Core ◽

Elliptical Model ◽

Long Time

The mechanisms of coherent tripole formation from unstable shielded circular vortices are analysed in the context of two-dimensional incompressible flows. Three stages are identified during the transformation process: the linear growth of the initial normal mode perturbation, its nonlinear amplification and the finite-amplitude saturation under the tripolar form. We give a geometrical discussion of the mutual influence of the core vortex and of the satellites generated from the shield. The role of the angular momentum in determining the finite amplitude saturation is demonstrated using a simple elliptical model of the core vortex associated with two point-vortex satellites. The long-time asymmetric breaking of the tripole into a dipole and a monopole is shown to be driven by the erosion of the core vortex by stripping and diffusion. Finally the influence of bottom topography on tripole formation is considered, providing a rich phenomenology when the height of the topography is varied.

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