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Bifurcation theory for vortices with application to boundary layer eruption

Published online by Cambridge University Press:  27 February 2019

Anne R. Nielsen Open the ORCID record for Anne R. Nielsen [Opens in a new window] ,
Matthias Heil Open the ORCID record for Matthias Heil [Opens in a new window] ,
Morten Andersen Open the ORCID record for Morten Andersen [Opens in a new window]  and
Morten Brøns Open the ORCID record for Morten Brøns [Opens in a new window]
Anne R. Nielsen
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Lyngby, Denmark
Matthias Heil
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Morten Andersen
Affiliation:
Department of Science and Environment, Roskilde University, 4000 Roskilde, Denmark
Morten Brøns*
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Lyngby, Denmark
*
†Email address for correspondence: [email protected]
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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 $Q$-criterion and analyse the vortex structure by considering the evolution of the zero contours of $Q$. 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 $Re=750$ to $Re=2250$. 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 $Re=1817$ 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.

JFM classification

Boundary Layers: Boundary layer structure Mathematical Foundations: Topological fluid dynamics Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 865 , 25 April 2019 , pp. 831 - 849
Copyright
© 2019 Cambridge University Press 

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