TY - CHAP

T1 - Linear and nonlinear instability of a vortex ring

AU - Fukumoto, Yasuhide

AU - Hattori, Yuji

PY - 2006/12/1

Y1 - 2006/12/1

N2 - A new linear instability mechanism of curvature origin is established for a vortex ring. The curvature effect reduces O(2) × SO(2) symmetry of a circularcylindrical tube to O(2), and fuels a pair of Kelvin waves whose azimuthal wavenumbers on the core are separated by one. For Kelvin's vortex ring, the growth rate and eigenfunctions are written out in closed form. In the inviscid case, the curvature effect dominates over the elliptically straining effect, but the former suffers from enhanced viscous damping. There are numerous excitable modes. As a first step toward an understanding of the route to a matured stage, we derive equations for weakly nonlinear evolution of amplitudes of the curvature instability. Our direct numerical simulation successfully captures the elliptical instability.

AB - A new linear instability mechanism of curvature origin is established for a vortex ring. The curvature effect reduces O(2) × SO(2) symmetry of a circularcylindrical tube to O(2), and fuels a pair of Kelvin waves whose azimuthal wavenumbers on the core are separated by one. For Kelvin's vortex ring, the growth rate and eigenfunctions are written out in closed form. In the inviscid case, the curvature effect dominates over the elliptically straining effect, but the former suffers from enhanced viscous damping. There are numerous excitable modes. As a first step toward an understanding of the route to a matured stage, we derive equations for weakly nonlinear evolution of amplitudes of the curvature instability. Our direct numerical simulation successfully captures the elliptical instability.

KW - Curvature instability

KW - Hamiltonian spectra

KW - Normal form

KW - Vortex ring

KW - Weakly nonlinear stability

UR - http://www.scopus.com/inward/record.url?scp=84859832868&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859832868&partnerID=8YFLogxK

M3 - Chapter

AN - SCOPUS:84859832868

SN - 9781402041808

T3 - Fluid Mechanics and its Applications

SP - 283

EP - 294

BT - IUTAM Symposium on Elementary Vortices and Coherent Structures

A2 - KIDA, SHIGEO

ER -