A dipole encounters a monopole
monopole weaker / encounter with offset
A Lamb dipole with radius 0.75 and
velocity of +2 (i.e. moving to the right) is placed to the
left of a positive Bessel monopole
with radius 0.50 at the origin.
In the main run of this example the
dipole is along the xaxis, making the encounter headon, and
the monpole has a circulation of +4, which means that the monopole is
weaker than each dipole half.
In this example:

The circulation of the monopole is +4, which makes the monopole
weaker
than each dipole half, since the circulation per dipole half is about 10.2.

The dipole is located 0.5 above the xaxis:
the encounter is with an offset.
The initial situation is shown in the following two graphs:


Initial tracer (left) and vorticity (right) distribution.
The domain shown is x=[3.0:+3.0], y=[2.0:+2.0].
The extrema of voriticity are +/ 29.5 for the dipole
and +11.8 for the monopole.
The monopole has an angular momentum: it rotates about its central point,
without moving.
The dipole has a linear momentum: it moves in the direction of its axis, in
this case to the right, towards the (weaker) monopole.
The monopole rotates counterclockwise, like the top half of the dipole.
The result of the rotation is that the dipole is pushed down, to negative
yvalues, while it moves towards the monopole, like in the
main run without offset. But since the
dipole now starts at a higher initial yvalue, the dipole reaches the
monopole with its negative half, rather than with its positive half.
This can be seen clearest in the tracer plots:

The negative dipole half (blue) then forms with the positive
monopole (green) a dipolar structure.
Since the monopole is weaker than the dipole half, the newly formed dipolar
structure will move along a curved path, with the stonger half on the
inside.
Meanwhile the original positive dipole half (red) is not a monopole, which
does not move on its own (there is a small motion, due to the dipolar
structure close by):


The curved path of the dipolar structure brings the negative dipole half
back close to its former positive and equallystrong partner.
Their combination is so strong that the negative dipole half teams up again
with its former partner to form a dipole. This dipole then moves away again
from the weaker positive monopole. Due to the interaction, however, there is
some shedding for vorticity and the monopole (green) has a small patch of
negative vorticity (blue) from the dipole.



In the main run without offset the monopole
merges with the dipole into a new, asymmetric dipole.
In this example there is an "exchange of partners", but this is only
temporarily, because the monopole is weaker than the dipole.
If this simulation is done
with a monopole that is nearly as strong as the
dipole, then the "exchange of partners" is permanent: the dipole moves
away to negative y, leaving the original positive dipole half (red)
behind as a monopole.

These MPEG movies (101 frames; 568kb) show the evolution until T=2.5
more clearly:

===>
MPEG movie of the tracer distribution

===>
MPEG movie of the vorticity distribution
At initialisation, a single tracer particle was placed at the extrema of
vorticity of both vortices. The next graph shows their trajectories:


Trajectories of the tracers, initially placed at
the extrema of vorticity of the monopole (solid
line) and the dipole (dashed line).
Symbols are placed at intervals of T=0.25.
Some further remarks
See the main run of this example,
with a weaker monopole and a headon encounter.
The evolution of the vorticity distribution is computed with a
Finite Difference Method
which solves the twodimensional vorticity (NavierStokes) equation.
Time and distances are given in dimensionless units.
===> Some details on the computation presented
on this page for those who are interested.
<=== Numerical simulations of 2D vortex
evolution with a Finite Difference Method.
Jos van Geffen 
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created: 20 September 1999
last modified: 26 May 2001