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8 - Interacting vortex pairs in inviscid and viscous planar flows

Published online by Cambridge University Press:  05 November 2012

By
T. Gallay
Edited by
James C. Robinson ,
José L. Rodrigo  and
Witold Sadowski
T. Gallay
Affiliation:
Université de Grenoble I
James C. Robinson
Affiliation:
University of Warwick
José L. Rodrigo
Affiliation:
University of Warwick
Witold Sadowski
Affiliation:
Uniwersytet Warszawski, Poland
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Summary

Abstract 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 vanishing viscosity limit, of the solution of the two dimensional Navier-Stokes equation with point vortices as initial data. In such a situation it is known, see Gallay (Arch. Ration. Mech. Anal.200 (2011) 445–490), that the solution behaves to leading order like a linear superposition of Oseen vortices whose centres evolve according to the point vortex system. However, higher order corrections can also be computed and these describe the deformation of the vortex cores due to mutual interactions. The second result is the construction by Smets & Van Schaftingen (Arch. Ration. Mech. Anal.198 (2010) 869–925) of “desingularized” solutions of the two-dimensional Euler equation. These solutions are stationary in a uniformly rotating or translating frame, and converge either to a single vortex or to a vortex pair as the size parameter ∈ goes to zero. We consider here the particular case of a pair of identical vortices, and we show that the solution of the weakly viscous Navier-Stokes equation is accurately described at time t by an approximate steady state of the rotating Euler equation which is a desingularized solution in the sense of Smets & Van Schaftingen (2010) with Gaussian profile and size

Introduction

Numerical simulations of freely decaying turbulence show that vortex interactions play a crucial role in the dynamics of two-dimensional viscous flows, see McWilliams (1984, 1990).

Type
Chapter
Information
Mathematical Aspects of Fluid Mechanics , pp. 173 - 200
Publisher: Cambridge University Press
Print publication year: 2012

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References

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