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Journal Articles Physics of Fluids Year : 2009

Three-dimensional instabilities and transient growth of a counter-rotating vortex pair


This paper investigates the three-dimensional instabilities and the transient growth of perturbations on a counter-rotating vortex pair. The two dimensional base flow is obtained by a direct numerical simulation initialized by two Lamb-Oseen vortices that quickly adjust to a flow with elliptic vortices. In the present study, the Reynolds number,ReΓ=Γ/ν, with Γ the circulation of one vortex and ν the kinematic viscosity, is taken large enough for the quasi steady assumption to be valid. Both the direct linearized Navier-Stokes equation and its adjoint are solved numerically and used to investigate transient and long time dynamics. The transient dynamics is led by different regions of the flow, depending on the optimal time considered. At very short times compared to the advection time of the dipole, the dynamics is concentrated on the points of maximal strain of the base flow, located at the periphery of the vortex core. At intermediate times, depending on the symmetry of the perturbation, one of the hyperbolic stagnation points provides the optimal amplification by stretching of the perturbation vorticity as in the classical hyperbolic instability. The growth of both short time and intermediate time transient perturbations are non- or weakly dependent of the axial wavenumber whereas the long time behavior strongly selects narrow bands of wavenumbers. We show that, for all unstable spanwise wavenumbers, the transient dynamics last until the nondimensional time t=2, during which the dipole has traveled twice the separation distance between vorticesb. During that time, all the wavenumbers exhibit a transient growth of energy by a factor of 50, for the Reynolds numberReΓ=2000. For time larger than t=2, energy starts growing at a rate given by the standard temporal stability theory. For all wavenumbers and two Reynolds numbers,ReΓ=2000 and ReΓ=105, different instability branches have been computed using a high resolution Krylov method. At large Reynolds number, the computed Crow and elliptic instability branches are in excellent agreement with the inviscid theory [S. C. Crow, AIAA J.8, 2172 (1970); S. Le Dizes and F. Laporte, J. Fluid Mech.471, 120 (2002)] and numerical analysis [D. Sipp and L. Jacquin, Phys. Fluids15, 1861 (2003)]. A novel oscillatory elliptic instability involving Kelvin waves with azimuthal wavenumbers m=0 and |m|=2, that was missed in previous numerical analysis [D. Sipp and L. Jacquin, Phys. Fluids15, 1861 (2003)] is found. For the stationary elliptic instability, we show that viscous effect may be estimated using the large Reynolds number direct and adjoint eigenmodes. This asymptotically exact estimate of the viscous damping of elliptic instability mode agrees with our direct numerical computation of instability branches at moderate Reynolds number and demonstrates that formula proposed by Le Dizes and Laporte [J. Fluid Mech.471, 120 (2002)] strongly over estimated the viscous correction.
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hal-01025605 , version 1 (18-07-2014)



Claire Donnadieu, Sabine Ortiz, Jean-Marc Chomaz, Paul Billant. Three-dimensional instabilities and transient growth of a counter-rotating vortex pair. Physics of Fluids, 2009, 21 (9), pp.094102. ⟨10.1063/1.3220173⟩. ⟨hal-01025605⟩
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