Abstracts - KTH Mechanics

The stability of multiple wing-tip vortices
E. J. Whitehead ∗ ,J.P.Denier ∗ andS.M.Cox ∗
The stability of wing-tip vortices is a problem that has received much attention
in the past 20 years due to their importance in the aerodynamics industry, where
they are the limiting factor that sets minimum spacing between following aircraft,
both in flight and during take-off and landing. Work to date can be divided into
two, not necessarily distinct, camps. The first approach follows the early work of
Crow 1 in which the stability of a vortex pair is considered, the resulting instability
now commonly referred to as the Crow instability. The second approach focus on the
stability of a single vortex, which is typically taken to be given by the asymptotic
solution for a single tip-vortex far downstream of the trailing edge, as was first derived
by Batchelor 2 .
Unlike much of the previous work on the stability of tip vortices we have considered
the stability of a multi-vortex system. The combined vortex flow is taken
as a combination of Batchelor vortices, who strengths, extents and positions can be
adjusted. Our results indicate that the optimal position, in terms of the maximal
streamwise growth rate of an infinitesimally small disturbance, for a four-vortex configuration
has all vortices aligned along the axis representing the aerofoil trailing edge;
see the accompanying figure. The horizontal locations of the vortices along this axis
that yield the maximal streamwise growth rate as a function of the relative vortex
strengths will also be described. If time permits, recent work on the question of the
absolute instability of multi-vortex flows will also be presented.
∗ School of Mathematical Sciences, The University of Adelaide, South Australia 5005, Australia.
1 Crow, AIAA J., 8, 2172 (1970).
2 Batchelor, J. Fluid Mech., 20, 645 (1964).
z−axis
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Figure 1: Contour plot of the maximum growth rate as a function of secondaryvortex
position. The vortices are aligned symmetrically with respect to the z-axis.
The primary vortex is situated at (y, z) =(2, 0).
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