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

Abstract

International audienceThis 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

Transient perturbation growth in time-dependent mixing layers

“…(For time-dependent equations, this can be directly shown by taking the adjoint of the propagator as given in equation (4) of Farrell & Ioannou 1996.) It is straightforward to establish (see for example Hill 1995;Donnadieu et al 2009) that the adjoint equations of (2.1) are…”

“…Throughout this paper we will set the central flow parameter Re = 1000 (except in § 5.4, where we study the effect of variations in Re), which is sufficiently large to ensure that the primary instability rolls up into a finite-amplitude, energetic billow. These adjoint equations can be solved using a straightforward modification of a pseudo-spectral DNS code with an Adams-Bashforth time scheme (as previously described and utilized by Donnadieu et al 2009). In particular, the central point arising from the integration by parts that the Laplacian operator has an opposite sign (and hence that the adjoint propagator is well-posed when integrating backward in time) is very simple to implement computationally.…”

“…It could seem that the abrupt change of slope of G max at the end of the peaked area around k z ≃ 1 is due to the cross-over of branches corresponding to different modes of the composed directadjoint propagator (Φ † λ Φ λ in § 2). A Krylov method similar to that described in Donnadieu et al (2009) was used to approximate the leading subdominant modes of the direct-adjoint propagator for T = 60. (The convergence of the results was checked by using different Krylov subspaces typically of dimension five or six.)…”

Control of Afterbody Vortices by Blowing

“…[14][15][16] It has been theoretically shown that three-dimensional instabilities may grow in a counter-rotating vortex pair. 17 The aims of this experimental study are to investigate the interactions between a pair of jets, emanating from the upswept face of a simplified fuselage model, and the afterbody vortex pair, and to understand how these interactions are affected by the blowing location and direction. This is compared against the baseline flow field (without blowing).…”

Optimal transient growth on a vortex ring and its transition via cascade of ringlets

“…Thin arrowed lines in (a) and (d) represent streamlines of the optimal perturbation and its outcome, respectively. (Moore & Saffman 1975;Donnadieu et al 2009). In the absence of axial base flow as in the present case, the elliptic instability was reported to be dominant over the curvature instability in a vortex ring flow (Blanco-Rodríguez & Le Dizèz 2016a,b).…”