## Motion of a two-dimensional monopolar vortex
in a bounded rectangular domain

J.H.G.M. van Geffen, V.V. Meleshko and G.J.F. van Heijst

*Physics of Fluids* **8**, 2393-2399 (1996)

### V. Conclusions

A distributed two-dimensional monopolar vortex in a bounded domain with
free-slip walls moves along the walls of the domain in a manner strikingly
similar to the motion of a point vortex of (initially) the same intensity
and starting at the same position - in spite of a considerable spreading of
the monopole's initial vorticity due to viscosity. This is independent of
the precise initial vorticity distribution of the monopole. Thus, the
(relatively) simple model of point vortices appears to be rather powerful in
describing the main features of the general behaviour during one revolution
of distributed vortex structures in a bounded domain with free-slip
boundaries.
After the first revolution a point vortex continues along the same
trajectory. The motion of the distributed monopole depends on viscous
effects: viscosity spreads the vorticity over a larger area and thus slows
down the monopole, which will thus spiral towards the centre of the domain.
The higher the Reynolds number is, the longer the monopole remains close to
the trajectory of the point vortex. For low Reynolds numbers the viscous
decay spreads the vorticity throughout the domain before the monopole has
completed one revolution.

In case of a domain with no-slip walls, *i.e.* zero velocities at the
walls, the distributed positive monopole moves along the wall and
immediately away from it along a curved but not completely smooth path to
the centre of the domain as a result of the negative vorticity induced by
the walls. The interaction between the positive and negative vorticity
causes (sometimes quite sharp) kinks to appear in the trajectory of the
monopoles: A large area of positive vorticity - rotating and moving due to
the walls - is surrounded by an asymmetric distribution of negative
vorticity. This combination acts like a kind of asymmetric dipolar vortex
and moves along a curved path, which bends if the negative vorticity area
breaks up and is spread around the positive vorticity. The closer the
monopole initially is to a boundary, the more negative vorticity is induced
near the walls and pulled into the domain and wrapped around the positive
monopole as it moves towards the centre, and the more important is the
precise initial vorticity distribution for the trajectory of the monopole's
centre.

Due to viscous diffusion, the relationship between vorticity and
streamfunction becomes well-defined when the monopole has reached the centre
of the domain, which implies that the resulting vorticity distribution is
axisymmetric.

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###### last modified: 26 May 2001