Bifurcation theory for vortices with application to boundary layer eruption

Anne R. Nielsen, Matthias Heil, Morten Andersen, Morten Brons*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

We develop a bifurcation theory describing the conditions under which vortices are created or destroyed in a two-dimensional incompressible flow. We define vortices using the -criterion and analyse the vortex structure by considering the evolution of the zero contours of. The theory identifies topological changes of the vortex structure and classifies these as four possible types of bifurcations, two occurring away from boundaries, and two occurring near no-slip walls. Our theory provides a description of all possible codimension-one bifurcations where time is treated as the bifurcation parameter. To illustrate our results, we consider the early stages of boundary layer eruption at moderate Reynolds numbers in the range from to. By analysing numerical simulations of the phenomenon, we show how to describe the eruption process as sequences of the four possible bifurcations of codimension one. Our simulations show that there is a single codimension-two point within our parameter range. This codimension-two point arises at via the coalescence of two codimension-one bifurcations which are associated with the creation and subsequent destruction of one of the vortices that erupt from the boundary layer. We present a theoretical description of this process and explain how the occurrence of this phenomenon separates the parameter space into two regions with distinct evolution of the topology of the vortices.

Original languageEnglish
JournalJournal of Fluid Mechanics
Volume865
Pages (from-to)831-849
ISSN0022-1120
DOIs
Publication statusPublished - 2019

Keywords

  • Boundary layer structure
  • Topological fluid dynamics
  • Vortex dynamics

Cite this

@article{6468f52e25e1448d821cd64e4c30f860,
title = "Bifurcation theory for vortices with application to boundary layer eruption",
abstract = "We develop a bifurcation theory describing the conditions under which vortices are created or destroyed in a two-dimensional incompressible flow. We define vortices using the -criterion and analyse the vortex structure by considering the evolution of the zero contours of. The theory identifies topological changes of the vortex structure and classifies these as four possible types of bifurcations, two occurring away from boundaries, and two occurring near no-slip walls. Our theory provides a description of all possible codimension-one bifurcations where time is treated as the bifurcation parameter. To illustrate our results, we consider the early stages of boundary layer eruption at moderate Reynolds numbers in the range from to. By analysing numerical simulations of the phenomenon, we show how to describe the eruption process as sequences of the four possible bifurcations of codimension one. Our simulations show that there is a single codimension-two point within our parameter range. This codimension-two point arises at via the coalescence of two codimension-one bifurcations which are associated with the creation and subsequent destruction of one of the vortices that erupt from the boundary layer. We present a theoretical description of this process and explain how the occurrence of this phenomenon separates the parameter space into two regions with distinct evolution of the topology of the vortices.",
keywords = "Boundary layer structure, Topological fluid dynamics, Vortex dynamics",
author = "Nielsen, {Anne R.} and Matthias Heil and Morten Andersen and Morten Brons",
year = "2019",
doi = "10.1017/jfm.2019.97",
language = "English",
volume = "865",
pages = "831--849",
journal = "Journal of Fluid Mechanics",
issn = "0022-1120",
publisher = "Cambridge University Press",

}

Bifurcation theory for vortices with application to boundary layer eruption. / Nielsen, Anne R.; Heil, Matthias; Andersen, Morten; Brons, Morten.

In: Journal of Fluid Mechanics, Vol. 865, 2019, p. 831-849.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Bifurcation theory for vortices with application to boundary layer eruption

AU - Nielsen, Anne R.

AU - Heil, Matthias

AU - Andersen, Morten

AU - Brons, Morten

PY - 2019

Y1 - 2019

N2 - We develop a bifurcation theory describing the conditions under which vortices are created or destroyed in a two-dimensional incompressible flow. We define vortices using the -criterion and analyse the vortex structure by considering the evolution of the zero contours of. The theory identifies topological changes of the vortex structure and classifies these as four possible types of bifurcations, two occurring away from boundaries, and two occurring near no-slip walls. Our theory provides a description of all possible codimension-one bifurcations where time is treated as the bifurcation parameter. To illustrate our results, we consider the early stages of boundary layer eruption at moderate Reynolds numbers in the range from to. By analysing numerical simulations of the phenomenon, we show how to describe the eruption process as sequences of the four possible bifurcations of codimension one. Our simulations show that there is a single codimension-two point within our parameter range. This codimension-two point arises at via the coalescence of two codimension-one bifurcations which are associated with the creation and subsequent destruction of one of the vortices that erupt from the boundary layer. We present a theoretical description of this process and explain how the occurrence of this phenomenon separates the parameter space into two regions with distinct evolution of the topology of the vortices.

AB - We develop a bifurcation theory describing the conditions under which vortices are created or destroyed in a two-dimensional incompressible flow. We define vortices using the -criterion and analyse the vortex structure by considering the evolution of the zero contours of. The theory identifies topological changes of the vortex structure and classifies these as four possible types of bifurcations, two occurring away from boundaries, and two occurring near no-slip walls. Our theory provides a description of all possible codimension-one bifurcations where time is treated as the bifurcation parameter. To illustrate our results, we consider the early stages of boundary layer eruption at moderate Reynolds numbers in the range from to. By analysing numerical simulations of the phenomenon, we show how to describe the eruption process as sequences of the four possible bifurcations of codimension one. Our simulations show that there is a single codimension-two point within our parameter range. This codimension-two point arises at via the coalescence of two codimension-one bifurcations which are associated with the creation and subsequent destruction of one of the vortices that erupt from the boundary layer. We present a theoretical description of this process and explain how the occurrence of this phenomenon separates the parameter space into two regions with distinct evolution of the topology of the vortices.

KW - Boundary layer structure

KW - Topological fluid dynamics

KW - Vortex dynamics

U2 - 10.1017/jfm.2019.97

DO - 10.1017/jfm.2019.97

M3 - Journal article

VL - 865

SP - 831

EP - 849

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -