Instabilities in co-rotating vortices with axial flow
Résumé
The wake shed by an aircraft is initially composed of several co- and counterrotatingvortices with axial jet whose strength and number depend on the spanwiseload distribution on the wings. For instance, a wing equipped with a single flapgenerates in the near field a wing tip and a co-rotating outboard flap tip vortex.When axial flow is not taken into account, this system of two vortices is known to beunstable with respect to a short-wavelength elliptical instability.1 This instability hasbeen shown to strongly affect the merging process of the vortices2 and its subsequentdynamics. In the present work, our goal is to analyze the effect of the axial flow onthe instability and the dynamics of the vortex pair.We consider the three-dimensional temporal evolution of two interacting parallelBatchelor vortices (in the regime where each vortex is inviscidly stable). From atheoretical point-of-view, each vortex is considered separately and the effect of theother vortex is modeled by a rotating strain field. In this framework, the ellipticinstability results from the resonant coupling of two Kelvin waves of the vortex withthe rotating strain field. Without axial flow, the most unstable mode is known tobe a combination of two helical Kelvin waves of azimuthal wavenumber m = −1and m = 1 leading to a sinuous deformation of each vortex. When axial flow isconsidered, the symmetry between left and right propagating waves is broken suchthat the combination of the two helical waves m = −1 and m = 1 is no longera sinuous deformation. In addition, one of the two waves becomes damped by theappearance of a critical layer such that the resonance between these two waves issuppressed above an axial flow threshold. However, the elliptical instability does notdisappear: other combinations of Kelvin waves m = −2 and m = 0, then m = −3and m = −1 are shown to become progressively unstable as axial flow is increased.A theoretical model for the instability growth rate is proposed. It is based on thelocal estimate of the elliptical instability growth rate in the vortex center.3 It also usesa large wavenumber asymptotic analysis4 in order to provide the characteristics of theresonant waves and an estimate of the damping rate associated with the critical layers.The theory is validated by the numerics. Temporal growth rates and unstable modestructures are shown to be well-reproduced by numerical simulations of the linearizedperturbation equations for a Batchelor vortex pair. Direct numerical simulations ofthe complete Navier-Stokes equations are also performed to analyze the nonlineardevelopment of the instability. The influence of the instability on the merging processand its impact on the global dynamics of the vortex system in the aeronautical contextare discussed.