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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