Interaction of Vortices in Weakly Viscous Planar Flows

@article{Gallay2011InteractionOV,
  title={Interaction of Vortices in Weakly Viscous Planar Flows},
  author={Thierry Gallay},
  journal={Archive for Rational Mechanics and Analysis},
  year={2011},
  volume={200},
  pages={445-490}
}
  • T. Gallay
  • Published 18 August 2009
  • Mathematics
  • Archive for Rational Mechanics and Analysis
We consider the inviscid limit for the two-dimensional incompressible Navier–Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter ν, and we choose a time T > 0 such that the Helmholtz–Kirchhoff point vortex system is well-posed on the interval [0, T]. Under these assumptions, we prove that the solution of the Navier–Stokes… 
Stability and Interaction of Vortices in Two-Dimensional Viscous Flows
The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the
Propagation of chaos for the 2D viscous vortex model
We consider a stochastic system of $N$ particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of
Interactions of three viscous point vortices
The dynamics of viscous point vortices in two dimensions is studied both analytically and numerically. We consider a core-growth model based on the Lamb–Oseen vortices, the so-called multi-Gaussian
Dynamics of several rigid bodies in a two-dimensional ideal fluid and convergence to vortex systems
We consider the motion of several solids in a bounded cavity filled with a perfect incompressible fluid, in two dimensions. The solids move according to Newton's law, under the influence of the
Dynamics of a point vortex as limits of a shrinking solid in an irrotational fluid
We consider the motion of a rigid body immersed in a two-dimensional perfect fluid. The fluid is assumed to be irrotational and confined in a bounded domain. We prove that when the body shrinks to a
Enhanced Dissipation and Axisymmetrization of Two-Dimensional Viscous Vortices
  • T. Gallay
  • Physics
    Archive for Rational Mechanics and Analysis
  • 2018
AbstractThis paper is devoted to the stability analysis of the Lamb–Oseen vortex in the regime of high circulation Reynolds numbers. When strongly localized perturbations are applied, it is shown
Concentrated Euler flows and point vortex model
This paper is an improvement of previous results on concentrated Euler flows and their connection with the point vortex model. Precisely, we study the time evolution of an incompressible two
Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid
The point vortex system is usually considered as an idealized model where the vorticity of an ideal incompressible two-dimensional fluid is concentrated in a finite number of moving points. In the
On the dynamics of vortices in viscous 2D flows
We study the 2D Navier–Stokes solution starting from an initial vorticity mildly concentrated near N distinct points in the plane. We prove quantitative estimates on the propagation of concentration
Existence and stability of viscous vortices
Vorticity plays a prominent role in the dynamics of incompressible viscous flows. In two-dimensional freely decaying turbulence, after a short transient period, evolution is essentially driven by
...
...

References

SHOWING 1-10 OF 60 REFERENCES
Interacting vortex pairs in inviscid and viscous planar flows
The aim of this contribution is to make a connection between two recent results concerning the dynamics of vortices in incompressible planar flows. The first one is an asymptotic expansion, in the
Generalized Helmholtz-Kirchhoff Model for Two-Dimensional Distributed Vortex Motion
TLDR
The two-dimensional Navier–Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations and it is proved that the convergence of this expansion is convergence and the Helmholtz–Kirchhoff model for the evolution of point vortices is recovered.
Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation
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
Spectral Properties of the Linearization at the Burgers Vortex in the High Rotation Limit
We study a linearized operator of the equation for the axisymmetric Burgers vortex which gives a stationary solution to the three dimensional Navier–Stokes equations with an axisymmetric background
Stretched vortices – the sinews of turbulence; large-Reynolds-number asymptotics
A large-Reynolds-number asymptotic theory is presented for the problem of a vortex tube of finite circulation [Gcy ] subjected to uniform non-axisymmetric irrotational strain, and aligned along an
Viscous interactions of two co-rotating vortices before merging
The viscous evolution of two co-rotating vortices is analysed using direct two-dimensional numerical simulations of the Navier–Stokes equations. The article focuses on vortex interaction regimes
Motion and Decay of a Vortex in a Nonuniform Stream
The motion of a vortex in a two‐dimensional incompressible nonuniform stream is studied by including the viscous effects in the inner core of the vortex. A systematic procedure is presented by the
Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity
Abstract.We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L1(R2) for positive times is entirely determined by the trace of
Uniqueness Theorem for the Basic Nonstationary Problem in the Dynamics of an Ideal Incompressible Fluid
A bstract . The initial boundary value problem is considered for the Euler equations for an incompressible fluid in a bounded domain D ⊂ Rn. It is known [Y1] that uniqueness holds for those flows
The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in ₃
It is shown here that a unique solution to the Navier-Stokes equations exists in R3 for a small time interval independent of the viscosity and that the solutions for varying viscosities converge
...
...