Numerical evidence of nonuniqueness in the evolution of vortex sheets

Milton C. Lopes Filho; John Lowengrub; Helena J. Nussenzveig Lopes; Yuxi Zheng

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 2, page 225-237
  • ISSN: 0764-583X

Abstract

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We consider a special configuration of vorticity that consists of a pair of externally tangent circular vortex sheets, each having a circularly symmetric core of bounded vorticity concentric to the sheet, and each core precisely balancing the vorticity mass of the sheet. This configuration is a stationary weak solution of the 2D incompressible Euler equations. We propose to perform numerical experiments to verify that certain approximations of this flow configuration converge to a non-stationary weak solution. Preliminary simulations presented here suggest this is indeed the case. We establish a convergence theorem for the vortex blob method that applies to this problem. This theorem and the preliminary calculations we carried out support the existence of two distinct weak solutions with the same initial data.


How to cite

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Lopes Filho, Milton C., et al. "Numerical evidence of nonuniqueness in the evolution of vortex sheets." ESAIM: Mathematical Modelling and Numerical Analysis 40.2 (2006): 225-237. <http://eudml.org/doc/249718>.

@article{LopesFilho2006,
abstract = {
We consider a special configuration of vorticity that consists of a pair of externally tangent circular vortex sheets, each having a circularly symmetric core of bounded vorticity concentric to the sheet, and each core precisely balancing the vorticity mass of the sheet. This configuration is a stationary weak solution of the 2D incompressible Euler equations. We propose to perform numerical experiments to verify that certain approximations of this flow configuration converge to a non-stationary weak solution. Preliminary simulations presented here suggest this is indeed the case. We establish a convergence theorem for the vortex blob method that applies to this problem. This theorem and the preliminary calculations we carried out support the existence of two distinct weak solutions with the same initial data.
},
author = {Lopes Filho, Milton C., Lowengrub, John, Nussenzveig Lopes, Helena J., Zheng, Yuxi},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonuniqueness; vortex sheets; vortex methods; Euler equations.; weak solution; incompressible Euler equations; convergence; vortex blob method},
language = {eng},
month = {6},
number = {2},
pages = {225-237},
publisher = {EDP Sciences},
title = {Numerical evidence of nonuniqueness in the evolution of vortex sheets},
url = {http://eudml.org/doc/249718},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Lopes Filho, Milton C.
AU - Lowengrub, John
AU - Nussenzveig Lopes, Helena J.
AU - Zheng, Yuxi
TI - Numerical evidence of nonuniqueness in the evolution of vortex sheets
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/6//
PB - EDP Sciences
VL - 40
IS - 2
SP - 225
EP - 237
AB - 
We consider a special configuration of vorticity that consists of a pair of externally tangent circular vortex sheets, each having a circularly symmetric core of bounded vorticity concentric to the sheet, and each core precisely balancing the vorticity mass of the sheet. This configuration is a stationary weak solution of the 2D incompressible Euler equations. We propose to perform numerical experiments to verify that certain approximations of this flow configuration converge to a non-stationary weak solution. Preliminary simulations presented here suggest this is indeed the case. We establish a convergence theorem for the vortex blob method that applies to this problem. This theorem and the preliminary calculations we carried out support the existence of two distinct weak solutions with the same initial data.

LA - eng
KW - Nonuniqueness; vortex sheets; vortex methods; Euler equations.; weak solution; incompressible Euler equations; convergence; vortex blob method
UR - http://eudml.org/doc/249718
ER -

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