resonant instability in two-dimensional vortex arrays by paolo luzzatto-fegiz, and charles h. k....
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Resonant instability in two-dimensional vortex arrays
by Paolo Luzzatto-Fegiz, and Charles H. K. Williamson
Proceedings AVolume 467(2128):1164-1185
April 8, 2011
©2011 by The Royal Society
Possible eigenvalue behaviour for (a) an exchange of stability (giving rise to a non-propagating unstable eigenmode), and (b) a Hamiltonian Hopf bifurcation (leading to an oscillatory
instability).
Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467:1164-1185
©2011 by The Royal Society
Schematic of perturbations involving (a) a displacement or (b) deformation of the vortex cores. m denotes the azimuthal wavenumber of the perturbation.
Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467:1164-1185
©2011 by The Royal Society
Perturbation involving an overall displacement of the configuration, schematically illustrated through the case of a co-rotating vortex pair.
Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467:1164-1185
©2011 by The Royal Society
Schematic of possible phase relations between disturbances on three vortices.
Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467:1164-1185
©2011 by The Royal Society
(a) Angular velocity for the elliptical model (dashed line) and the full solution (solid line) for three identical co-rotating vortices.
Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467:1164-1185
©2011 by The Royal Society
Selected eigenvalues for three co-rotating vortices, showing the prediction for the development of the first resonance from the elliptical model (a), together with results from an accurate linear
stability analysis (b).
Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467:1164-1185
©2011 by The Royal Society
Eigenvalues for three co-rotating vortices.
Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467:1164-1185
©2011 by The Royal Society
Close-up view of the second resonance occurring for three vortices.
Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467:1164-1185
©2011 by The Royal Society
Imaginary part of the overall-displacement eigenvalue σI and of the angular velocity Ω of the configuration.
Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467:1164-1185
©2011 by The Royal Society