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

  • Navrose (a1) (a2), H. G. Johnson (a1), V. Brion (a1), L. Jacquin (a1) and J. C. Robinet (a3)...


We investigate perturbations that maximize the gain of disturbance energy in a two-dimensional isolated vortex and a counter-rotating vortex pair. The optimization is carried out using the method of Lagrange multipliers. For low initial energy of the perturbation ( $E(0)$ ), the nonlinear optimal perturbation/gain is found to be the same as the linear optimal perturbation/gain. Beyond a certain threshold $E(0)$ , the optimal perturbation/gain obtained from linear and nonlinear computations are different. There exists a range of $E(0)$ for which the nonlinear optimal gain is higher than the linear optimal gain. For an isolated vortex, the higher value of nonlinear optimal gain is attributed to interaction among different azimuthal components, which is otherwise absent in a linearized system. Spiral dislocations are found in the nonlinear optimal perturbation at the radial location where the most dominant wavenumber changes. Long-time nonlinear evolution of linear and nonlinear optimal perturbations 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 time scale. The quasi-steady vortex state is non-axisymmetric and its shape depends on the initial perturbation. It is observed that the lifetime of a quasi-steady vortex state obtained using the nonlinear optimal perturbation is longer than that obtained using the linear optimal perturbation. For a 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, the optimal perturbation for a vortex pair can be either symmetric or antisymmetric, whereas the structure of the nonlinear optimal perturbation, beyond the threshold $E(0)$ , is always asymmetric. No quasi-steady state for a counter-rotating vortex pair is observed.


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Antkowiak, A. & Brancher, P. 2004 Transient energy growth for the Lamb–Oseen vortex. Phys. Fluids 16 (1), L1L4.
Antkowiak, A. & Brancher, P. 2007 On vortex rings around vortices: an optimal mechanism. J. Fluid Mech. 578, 295304.
Barkley, D. 2016 Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803, P1.
Bernoff, A. J. & Lingevitch, J. F. 1994 Rapid relaxation of an axisymmteric vortex. Phys. Fluids 6 (11), 37173723.
Bisanti, L.2013 Linear and nonlinear optimal perturbation analysis of vortices in incompressible flows. PhD Thesis, Institute National Polytechnique de Toulouse, Université de Toulouse.
Brion, V.2009 Stabilité de paires de tourbillons contra-rotatifs: application au tourbillon de jeu dans les turbomachines. PhD Thesis, École Poytechnique, Palaiseau.
Brion, V., Sipp, D. & Jacquin, L. 2014 Linear dynamics of the Lamb–Chaplygin dipole in the two-dimensional limit. Phys. Fluids 26 (6), 064103.
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2011 The minimal seed of turbulent transition in the boundary layer. J. Fluid Mech. 689, 221253.
Cherubini, S. & De Palma, P. 2013 Nonlinear optimal perturbations in a Couette flow: bursting and transition. J. Fluid Mech. 716, 251279.
Corbett, P. & Bottaro, A. 2000 Optimal perturbations for bondary layers subject to streamwise pressure gradient. Phys. Fluids 12, 120130.
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8 (12), 21722179.
Douglas, S. C., Amari, S.-I. & Kung, S.-Y. 2000 On gradient adaptation with unit-norm constraints. IEEE Trans. Signal Process. 48 (6), 18431847.
Fabre, D., Sipp, D. & Jacquin, L. 2006 Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235274.
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31 (8), 20932102.
Fischer, P., Lottes, J. & Kerkemeier, S.2008 Nek5000 webpage.
Green, S. I. 1995 Fluid Vortices. Kluwer Academic.
Habermann, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths 51 (2), 139161.
Jugier, R.2016 Stabilité bidimensionnelle de modeles de sillage d’aéronefs. PhD Thesis, Institut Supérieur de l’Aéronautique et de l’Espace, Université de Toulouse.
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10 (61), 155168.
Kerswell, R. R. 2018 Nonlinear nonmodal stability theory. Annu. Rev. Fluid Mech. 50 (1), 319345.
Küchemann, D. 1965 Report on the IUTAM symposium on concentrated vortex motions in fluids. J. Fluid Mech. 21 (1), 120.
Le Dizes, S. 2000 Non-axisymmetric vortices in two-dimensional flows. J. Fluid Mech. 406, 175198.
Lugt, H. J. 1983 Vortex Flow in Nature and Technology. Wiley.
Mao, X. & Sherwin, S. 2011 Continuous spectra of the Batchelor vortex. J. Fluid Mech. 681, 123.
Mao, X. & Sherwin, S. 2012 Transient growth associated with continuous spectra of the Batchelor vortex. J. Fluid Mech. 697, 3559.
Orr, W. McF. 1907 Stability or instability of the steady motions of a perfect liquid. Proc. Ir. Acad. Sect. A 27, 969.
Pierrehumbert, R. 1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99 (1), 129144.
Pradeep, D. S. & Hussain, F. 2006 Transient growth of perturbations in a vortex column. J. Fluid Mech. 550, 251288.
Rossi, L. F., Lingevitch, J. F. & Bernoff, A. J. 1997 Quasi-steady monopole and tripole attractors for relaxing vortices. Phys. Fluids 9 (8), 23292338.
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Schmid, P. J. & Brandt, L. 2014 Analysis of fluid systems: stability, receptivity, sensitivity. Lecture Notes from the FLOW-NORDITA Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013. Appl. Mech. Rev. 66 (2), 024803.
Sipp, D., Jacquin, L. & Cossu, C. 2000 Self-adaptation and viscous selection in concentrated two-dimensional vortex dipoles. Phys. Fluids 12 (2), 245248.
Trefethen, L., Trefethen, A., Reddy, S. & Driscoll, T. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578584.
Tsai, C.-Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73 (04), 721733.
Zuccher, S., Bottaro, A. & Luchini, P. 2006 Algebraic growth in a Blasius boundary layer: nonlinear optimal disturbances. Eur. J. Mech. (B/Fluids) 25, 117.
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Journal of Fluid Mechanics
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