Abstracts - KTH Mechanics

198
Instabilities in co-rotating vortices with axial flow
N. Schaeffer ∗ , L. Lacaze † , S. Le Dizès ∗
The wake shed by an aircraft is initially composed of several co- and counterrotating
vortices with axial jet whose strength and number depend on the spanwise
load distribution on the wings. For instance, a wing equipped with a single flap
generates 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 be
unstable with respect to a short-wavelength elliptical instability. 1 This instability has
been shown to strongly affect the merging process of the vortices 2 and its subsequent
dynamics. In the present work, our goal is to analyze the effect of the axial flow on
the instability and the dynamics of the vortex pair.
We consider the three-dimensional temporal evolution of two interacting parallel
Batchelor vortices (in the regime where each vortex is inviscidly stable). From a
theoretical point-of-view, each vortex is considered separately and the effect of the
other vortex is modeled by a rotating strain field. In this framework, the elliptic
instability results from the resonant coupling of two Kelvin waves of the vortex with
the rotating strain field. Without axial flow, the most unstable mode is known to
be a combination of two helical Kelvin waves of azimuthal wavenumber m = −1
and m = 1 leading to a sinuous deformation of each vortex. When axial flow is
considered, the symmetry between left and right propagating waves is broken such
that the combination of the two helical waves m = −1 andm = 1 is no longer
a sinuous deformation. In addition, one of the two waves becomes damped by the
appearance of a critical layer such that the resonance between these two waves is
suppressed above an axial flow threshold. However, the elliptical instability does not
disappear: other combinations of Kelvin waves m = −2 andm =0,thenm = −3
and 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 the
local estimate of the elliptical instability growth rate in the vortex center. 3 It also uses
a large wavenumber asymptotic analysis 4 in order to provide the characteristics of the
resonant 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 mode
structures are shown to be well-reproduced by numerical simulations of the linearized
perturbation equations for a Batchelor vortex pair. Direct numerical simulations of
the complete Navier-Stokes equations are also performed to analyze the nonlinear
development of the instability. The influence of the instability on the merging process
and its impact on the global dynamics of the vortex system in the aeronautical context
are discussed.
This work is supported by the European Community (FAR-Wake project) and the
French Agency for Research (ANR).
∗ IRPHE, 49 rue F. Joliot Curie, BP 146, F-13384 Marseille cedex 13, France.
† IMFT, 1 Allée du Professeur Camille Soula, F-31400 Toulouse, France.
1 Le Dizès and Laporte, J. Fluid Mech. 471, 169–201 (2002).
2 Meunier and Leweke, J. Fluid Mech. 533, 125-159 (2005).
3 Waleffe, Phys. Fluids A2, 76–80 (1990).
4 Le Dizès and Lacaze, J. Fluid Mech. 542, 69–96 (2005).