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Optimal perturbation for two-dimensional vortex systems: route to non-axisymmetric state

Abstract : We investigate perturbations that maximize the gain of disturbance energy in two-dimensional isolated vortex and counter-rotating vortex-pair. The optimization is carried out using the method of Lagrange multipliers. For low initial energy of the perturbation (E(0)), nonlinear optimal perturbation/gain is found to be the same as linear optimal perturbation/gain. Beyond a certain threshold E(0), optimal perturbation/gain obtained from linear and nonlinear computation is different. There exists a range of E(0) for which nonlinear optimal gain is higher than linear optimal gain. For isolated vortex, higher value of nonlinear optimal gain is attributed to interaction among different azimuthal components, that is otherwise absent in a linearized system. Spiral dislocations are found in nonlinear optimal perturbation at the radial location where the most dominant wavenumber changes. Long-time nonlinear evolution of linear and nonlinear optimal perturbation is studied. The evolution shows that after the initial increment of perturbation energy, the vortex attains a quasi-steady state where the mean perturbation energy decreases on a slow timescale. The quasi-steady vortex state is non-axisymmetric and its shape depends on the initial perturbation. It is observed that the life of a quasi-steady vortex state obtained using nonlinear optimal perturbation is longer than that obtained using linear optimal perturbation. For counter-rotating vortex-pair, the mechanism that maximizes the energy gain is found to be similar to that of the isolated vortex. Within the linear framework optimal perturbation for a vortex pair can be either symmetric or anti-symmetric, whereas, the structure of the nonlinear optimal perturbation, beyond the threshold E(0), is always asymmetric. Quasi-steady state for counter-rotating vortex pair is not observed.
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Submitted on : Friday, July 24, 2020 - 9:54:13 AM
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Navrose Navrose, H. Johnson, Vincent Brion, Laurent Jacquin, Jean-Christophe Robinet. Optimal perturbation for two-dimensional vortex systems: route to non-axisymmetric state. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2018, 855, pp.922-952. ⟨10.1017/jfm.2018.689⟩. ⟨hal-02906092⟩

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