Bifurcation theory for vortices with application to boundary layer eruption

Anne R. Nielsen, Matthias Heil, Morten Wienecke Andersen, Morten Brøns

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

Resumé

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.
OriginalsprogEngelsk
TidsskriftJournal of Fluid Mechanics
Vol/bind2019
Udgave nummer865
Sider (fra-til)831-849
Antal sider19
ISSN0022-1120
DOI
StatusUdgivet - 2019

Citer dette

Nielsen, Anne R. ; Heil, Matthias ; Wienecke Andersen, Morten ; Brøns, Morten. / Bifurcation theory for vortices with application to boundary layer eruption. I: Journal of Fluid Mechanics. 2019 ; Bind 2019, Nr. 865. s. 831-849.
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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.",
author = "Nielsen, {Anne R.} and Matthias Heil and {Wienecke Andersen}, Morten and Morten Br{\o}ns",
year = "2019",
doi = "10.1017/jfm.2019.97",
language = "English",
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Bifurcation theory for vortices with application to boundary layer eruption. / Nielsen, Anne R. ; Heil, Matthias; Wienecke Andersen, Morten; Brøns, Morten.

I: Journal of Fluid Mechanics, Bind 2019, Nr. 865, 2019, s. 831-849.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

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

AU - Nielsen, Anne R.

AU - Heil, Matthias

AU - Wienecke Andersen, Morten

AU - Brøns, 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.

U2 - 10.1017/jfm.2019.97

DO - 10.1017/jfm.2019.97

M3 - Journal article

VL - 2019

SP - 831

EP - 849

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

IS - 865

ER -