We study the nonlinear evolution of the elliptic instability and its influence on the merging process of two corotating Batchelor vortices using a spectral DNS approach. First, we analyse the nonlinear saturation of the elliptic instability for a single strained vortex, with and without axial jet, for moderate Reynolds numbers (Re= Gamma/nu approx= 12500, where Gamma is the circulation and nu the kinematic viscosity). We show that the vortex deformation induced by the instability remains limited to the vortex core region. The second part of our work focuses on the influence of the elliptic instability on the merging process. We compare three cases : no instability (2D), elliptic instability without axial jet, and elliptic instability with axial jet, the latter case being relevant to aircraft wakes. Qualitative and quantitative differences between the three different cases are pointed out and discussed in the context of aircraft vortices

LE DIZÈS, Stéphane

SCHAEFFER, Nathanaël

I-Revues

18èmeCongrès Français de Mécanique Grenoble, 27-31 août 2007
Influence of an elliptic instability on the merging of a co-rotating vortex pair
Nathanaël Schaeffer & Stéphane Le Dizès
CNRS
IRPHE - UMR 6594
B.P. 146 13384 Marseille Cedex 13, France
email of the corresponding author
Abstract :
We study the nonlinear evolution of the elliptic instability and its influence on the merging process of two corotating
Batchelor vortices using a spectral DNS approach.
First, we analyse the nonlinear saturation of the elliptic instability for a single strained vortex, with and without
axial jet, for moderate Reynolds numbers (Re = Γ/ν ≈ 12500, whereΓ is the circulation andν the kinematic
viscosity). We show that the vortex deformation induced by the instability remains limited to the vortex core region.
The second part of our work focuses on the influence of the elliptic instability on the merging process. We compare
three cases : no instability (2D), elliptic instability without axial jet, and elliptic instability with axial jet, thelatter
case being relevant to aircraft wakes. Qualitative and quantit tive differences between the three different cases are
pointed out and discussed in the context of aircraft vortices.
Résumé :
Nous étudions l’évolution non-linéaire de l’instabilité elliptique et son influence lors de la fusion de deux vortex
de Batchelor corotatifs, à l’aide de simulations numériques.
Dans un premier temps, nous analyson la saturation non-linéaire de l’instabilité elliptique pour un tourbillon dans
un champ d’étirement, avec et sans jet axial, à des nombres deReynolds modérés (Re = Γ/ν ≈ 12500, oùΓ est
la circulation etν la viscosité cinématique). Nous montrons que la déformation du vortex induite par l’instabilité
reste limitée au cœur.
La seconde partie de ce travail examine l’influence de l’instabilité elliptique lors du processus de fusion. Nous
comparons trois cas de fusion : sans instabilité (2D), avec l’instabilité elliptique sans jet axial, et avec l’instabilité
elliptique avec jet axial, ce dernier cas ayant des applications aux sillages des avions. Les différences qualitatives
et quantitatives entre ces trois cas sont examinées et discutées dans le contexte aéronautique.
Key-words :
vortex merging; elliptic instability
1 Introduction
A vortex elliptically deformed by an external strain field issubjected to the so-called elliptic
instability, resulting in the deformation of the vortex core, along the vortex axis. The linear
properties of this instability have been studied extensively, in various cases : co-rotating and
counter-rotating vortex pair, with or without axial jet. This work focuses on the non-linear
evolution of this instability and its effect on the merging of a co-rotating vortex pair.
The elliptic instability has been thoroughly reviewed by Kerswell (2002). It can be viewed
as a resonance between two Kelvin waves (or eigen modes, see Fabr et al., 2006) of the vortex
and the external shear. Theoretical predictions have been done by Le Dizès & Laporte (2002),
and a study of the instability of counter-rotating vorticeswith axial jet has been carried out
recently by Lacazeet al. (2007). Only little is known about the non-linear regime : aweakly
non-linear analysis has been done by Sipp (2000) and Meunier& L weke (2005) performed
some experiments on counter-rotating vortices without axial flow.
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18èmeCongrès Français de Mécanique Grenoble, 27-31 août 2007
Using numerical simulations on Batchelor vortices, we first study the saturation of the ellip-
tic instability on a single vortex, showing that the sinuousdisplacement induced by the elliptic
instability remains bounded. The second part of this work shows that a merging with elliptic
instability happens earlier and makes the resulting vortexwider.
2 Method
The initial,z-invariant, vorticity fieldω is composed of two vortices of sizea and circulationΓ,
separated by a distanceb :
ω =
Γ
πa2
(
exp
(
−(x − b/2)2/a2 − x2/a2
)
+ exp
(
−(x + b/2)2/a2 − x2/a2
))
The axial velocityvz in a Batchelor vortex is proportional to the vorticity and itsintensity is
measured by the parameterW0 = 2vz/ωa.
The vortex core sizea increases due to viscous diffusion, and whena/b reaches a threshold
value close to 0.23 in 2D, the merging process begins. This process is well-documented when
the system remains two-dimensional. When the Reynolds numberis large enough, each vortex
develops an elliptic instability, deforming the vortex core and eventually modifying the merging
process.
In order to tackle this problem, we use a full spectral method. The computation domain is a
square of sizeL with the origin at its center. In thez direction, a simple Fourier decomposition
is performed.
3 Non-linear evolution of the elliptic instability
For this part, we freeze the two-dimensional flow, and let theelliptic instability develop on top
of thez-invariant stationary flow. From previous studies it is knowthat without axial jet, the
most unstable mode is a combination ofm = 1 andm = −1, leading to a sinuous deformation.
The structure of the vortex perturbed by the elliptic instability is illustrated in figure 1.
We have observed that during the nonlinear evolution, the vorticity field gets more and more
localised until viscous effects stop this process. The casewith axial jet, is similar but the most
unstable mode is now a combination ofm = 2 andm = 0, leading to a helical deformation.
Figure 1: Vortex deformed by the elliptic instability materialized by a total vorticity isosurface, without
axial jet (top line) and withW0 = 0.6 (bottom line). A weak, almost linear deformation is shown on the
left, whereas a strongly non-linear case is shown on the right.
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18èmeCongrès Français de Mécanique Grenoble, 27-31 août 2007
4 Merging of a vortex pair subjected to elliptic instability
In order to compare how three different configurations of a vortex pair will merge, we start with
the two-dimensional flow described above, and let it evolve to reach a quasi-steady state. At a
given time, we add the most unstable mode at a given amplitude, as well as the optional axial
jet, and let the resulting three different system evolve further. The evolution of the distance
between the two vortices is shown in figure 2. The most unstable mode is added ett = 100
(time is based on the single vortex turn-over time) and we cansee that the merging occurs earlier
when an elliptic instability is allowed to develop, with or without jet.
100 120 140 160 180 200 220 240
3
3.5
4
4.5
5
t
b
merging at Re = 12500
2D
w/o jet
with jet (W0 = 0.6)
Figure 2: Distanceb between the two vortices as a function of time, atRe = 12500
When the merging process is completed and the three systems have reached a nearly ax-
isymmetric state, we compare the vorticity profiles (figure 3). When compared to the two-
dimensional case, the merging processes with elliptic instability evolve toward wider vortices,
with lower peak vorticity. The vortex widening is even stronger with axial jet. We found also
that the peak angular velocity is reduced by about 20%.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
0.2
0.4
0.6
0.8
1
1.2
r
ω
2D
w/o jet
with jet (W
0
=0.6)
Figure 3: Vorticity profiles after merging, taken at the same time, atRe = 12500
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18èmeCongrès Français de Mécanique Grenoble, 27-31 août 2007
5 Conclusions
The non-linear evolution of the elliptic instability has been investigated. For a single strained
vortex, the displacement of the vortex core remains bounded, even though the amplitude of the
instability can grow quite large, resulting in important deformations. This can be explained by
the presence of one or more zeros in the radial profile of the most unstable mode.
The merging of a co-rotating vortex pair is strongly affected by the elliptic instability : the
process happens earlier, and the resulting vortex is significa tly wider and weaker. For aircraft
wakes, this means that in tuning the jet and the circulation,it is possible to achieve a faster
decay of the strong trailing vortices.
References
Fabre, D., Sipp, D., Jacquin, L. 2006, The Kelvin waves and the singular modes of the
Lamb-Oseen vortex.J. Fluid Mech.551235-274
Kerswell, R.R. 2002, Elliptical instability.Annu. Rev. Fluid Mech.3483-113
Lacaze, L., Ryan, K., Le Dizès, S. 2007, Elliptic instabilityin a strained Batchelor vortexJ.
Fluid Mech.in press.
Le Dizès, S., Laporte, F. 2002, Theoretical predictions forthe elliptic instability in a two-vortex
flow J. Fluid Mech.471169-201
Meunier, P., Leweke, T. 2005, Elliptic instability of a co-rtating vortex pairJ. Fluid Mech.533
125-159
Sipp, D 2000, Weakly nonlinear saturation of short-wave instabilities in a strained Lamb-Oseen
vortex.Phys. FLuids121715-1729
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Influence of an elliptic instability on the merging of a co-rotating vortex pair

Schaeffer, Nathanael

Le Dizès, Stéphane

HAL AMU

http://documents.irevues.inist.fr/bitstream/2042/16720/1/CFM2007-1038.pdf

Influence of an elliptic instability on the merging of a co-rotating vortex pair

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