Lagrangian approach to weakly nonlinear stability of elliptical flow

Yasuhide Fukumoto, M. Hirota, Y. Mie

Research output: Contribution to journalConference article

5 Citations (Scopus)

Abstract

Rotating flows with elliptically strained streamlines suffer from parametric resonance instability between a pair of Kelvin waves whose azimuthal wavenumbers are separated by two. We address the weakly nonlinear amplitude evolution of three-dimensional (3D) Kelvin waves, in resonance, on a flow confined in a cylinder of elliptic cross-section. In a traditional Eulerian approach, derivation of the mean flow induced by nonlinear interaction of Kelvin waves stands as an obstacle. We show how a topological idea, or the Lagrangian approach, facilitates calculation of the wave-induced mean flow. A steady incompressible Euler flow is characterized as a state of the maximum of the total kinetic energy with respect to perturbations constrained to an isovortical sheet, and the isovortical perturbation is handled only in terms of the Lagrangian variables. The criticality in energy of a steady flow allows us to calculate the wave-induced mean flow only from the linear Lagrangian displacement. With the mean flow at hand, the Lagrangian approach provides us with a shortcut to enter into a weakly nonlinear amplitude evolution regime of 3D disturbances. Unlike the Eulerian approach, the amplitude equations are available directly in the Hamiltonian normal form.

Original languageEnglish
Article number014049
JournalPhysica Scripta T
VolumeT142
DOIs
Publication statusPublished - Dec 1 2010
Event2nd International Conference and Advanced School on Turbulent Mixing and Beyond, TMB-2009 - Trieste, Italy
Duration: Jul 27 2009Aug 7 2009

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Nonlinear Stability
Kelvin waves
Kelvin
Perturbation
Rotating Flow
Parametric Resonance
Amplitude Equations
Nonlinear Interaction
perturbation
Criticality
Streamlines
Steady Flow
Kinetic energy
steady flow
Normal Form
Euler
Cross section
Disturbance
disturbances
derivation

All Science Journal Classification (ASJC) codes

  • Atomic and Molecular Physics, and Optics
  • Mathematical Physics
  • Condensed Matter Physics
  • Physics and Astronomy(all)

Cite this

Lagrangian approach to weakly nonlinear stability of elliptical flow. / Fukumoto, Yasuhide; Hirota, M.; Mie, Y.

In: Physica Scripta T, Vol. T142, 014049, 01.12.2010.

Research output: Contribution to journalConference article

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