Nonlinear analysis for the evolution of vortex sheets*

Russel Caflisch

Research output: Contribution to journalArticle

Abstract

The perturbations in a nearly flat vortex sheet will initially grow due to Kelvin-Helmholtz instability. Asymptotic analysis and numerical computations of the subsequent nonlinear evolution show several interesting features. At some finite time the vortex sheet develops a singularity in its shape; i.e. the curvature becomes infinite at a point. This is immediately followed by roll-up of the sheet into an infinite spiral. This paper presents two mathematical results on nonlinear vortex sheet evolution and singularity formation: First, for sufficiently small analytic perturbations of the flat sheet, existence of smooth solutions of the Birkhoff-Rott equation is proved almost up to the expected time of singularity formation. Second, we present a construction of exact solutions that develop singularities (infinite curvature) in finite time starting from analytic initial data. These results are derived within the framework of analytic function theory.

Original languageEnglish (US)
Pages (from-to)75-77
Number of pages3
JournalFluid Dynamics Research
Volume3
Issue number1-4
DOIs
StatePublished - Sep 1 1988

Fingerprint

vortex sheets
Nonlinear analysis
Vortex flow
Asymptotic analysis
curvature
perturbation
analytic functions
Kelvin-Helmholtz instability

ASJC Scopus subject areas

  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Fluid Flow and Transfer Processes

Cite this

Nonlinear analysis for the evolution of vortex sheets*. / Caflisch, Russel.

In: Fluid Dynamics Research, Vol. 3, No. 1-4, 01.09.1988, p. 75-77.

Research output: Contribution to journalArticle

Caflisch, Russel. / Nonlinear analysis for the evolution of vortex sheets*. In: Fluid Dynamics Research. 1988 ; Vol. 3, No. 1-4. pp. 75-77.
@article{a7074011ebc04eb4b123c46d7a057e66,
title = "Nonlinear analysis for the evolution of vortex sheets*",
abstract = "The perturbations in a nearly flat vortex sheet will initially grow due to Kelvin-Helmholtz instability. Asymptotic analysis and numerical computations of the subsequent nonlinear evolution show several interesting features. At some finite time the vortex sheet develops a singularity in its shape; i.e. the curvature becomes infinite at a point. This is immediately followed by roll-up of the sheet into an infinite spiral. This paper presents two mathematical results on nonlinear vortex sheet evolution and singularity formation: First, for sufficiently small analytic perturbations of the flat sheet, existence of smooth solutions of the Birkhoff-Rott equation is proved almost up to the expected time of singularity formation. Second, we present a construction of exact solutions that develop singularities (infinite curvature) in finite time starting from analytic initial data. These results are derived within the framework of analytic function theory.",
author = "Russel Caflisch",
year = "1988",
month = "9",
day = "1",
doi = "10.1016/0169-5983(88)90045-7",
language = "English (US)",
volume = "3",
pages = "75--77",
journal = "Fluid Dynamics Research",
issn = "0169-5983",
publisher = "IOP Publishing Ltd.",
number = "1-4",

}

TY - JOUR

T1 - Nonlinear analysis for the evolution of vortex sheets*

AU - Caflisch, Russel

PY - 1988/9/1

Y1 - 1988/9/1

N2 - The perturbations in a nearly flat vortex sheet will initially grow due to Kelvin-Helmholtz instability. Asymptotic analysis and numerical computations of the subsequent nonlinear evolution show several interesting features. At some finite time the vortex sheet develops a singularity in its shape; i.e. the curvature becomes infinite at a point. This is immediately followed by roll-up of the sheet into an infinite spiral. This paper presents two mathematical results on nonlinear vortex sheet evolution and singularity formation: First, for sufficiently small analytic perturbations of the flat sheet, existence of smooth solutions of the Birkhoff-Rott equation is proved almost up to the expected time of singularity formation. Second, we present a construction of exact solutions that develop singularities (infinite curvature) in finite time starting from analytic initial data. These results are derived within the framework of analytic function theory.

AB - The perturbations in a nearly flat vortex sheet will initially grow due to Kelvin-Helmholtz instability. Asymptotic analysis and numerical computations of the subsequent nonlinear evolution show several interesting features. At some finite time the vortex sheet develops a singularity in its shape; i.e. the curvature becomes infinite at a point. This is immediately followed by roll-up of the sheet into an infinite spiral. This paper presents two mathematical results on nonlinear vortex sheet evolution and singularity formation: First, for sufficiently small analytic perturbations of the flat sheet, existence of smooth solutions of the Birkhoff-Rott equation is proved almost up to the expected time of singularity formation. Second, we present a construction of exact solutions that develop singularities (infinite curvature) in finite time starting from analytic initial data. These results are derived within the framework of analytic function theory.

UR - http://www.scopus.com/inward/record.url?scp=0024067931&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0024067931&partnerID=8YFLogxK

U2 - 10.1016/0169-5983(88)90045-7

DO - 10.1016/0169-5983(88)90045-7

M3 - Article

VL - 3

SP - 75

EP - 77

JO - Fluid Dynamics Research

JF - Fluid Dynamics Research

SN - 0169-5983

IS - 1-4

ER -