## Interaction of 16 monopoles - not symmetrical with stress-free walls

An example of self-organisation of two-dimensional flows

In this simulation 16 single signed monopolar vortices, so-called Bessel monopoles, interact with each other in a square domain with stress-free walls all around. Initially they are placed not exactly symmetric to make things more interesting.

Initial vorticity distribution: 8 positive and 8 negative Bessel monopoles of radius 0.45 are placed around the centre a 7 by 7 domain, but not exactly symmetric. (You can click on the pictures for a larger version.)

The boundary condition of stress-free walls implies that fluid cannot go through the walls, but can move freely along them. In physical terms: the velocity of the fluid perpendicular to the wall is zero, whereas the velocity tangential to the wall is undetermined. A vortex reaching such a boundary sort of 'feels' its reflection in the wall and interacts with it. The motion of each monopole is therefore determined by the 15 other monopoles and by the 'reflections' of all monopoles in the walls.

Vorticity distribution at t=5 (left) and t=10 (right).

Apparently some monopoles join with oppositely signed monopoles and form dipolar structures, while other monopoles are torn apart and merge with monopoles of the same sign to form larger monopoles. Small scale features gradually disappear and there is a tendency for the vorticity to go to larger scales. This proces is called self-organisation of two-dimensional flows. Since the initial state is not symmetrical, the evolution is somewhat chaotic.

During the evolution viscosity spreads the vorticity over a larger area and reduces the extrema of vorticity, as a result of which the monopoles move and rotate slower. The colours in the pictures stand for the same vorticity levels throughout the evoltuion shown in the pictures. The reduction of the vorticity extrema is therefore visible as a disappearing of colours as time goes on.

Vorticity distribution at t=25 (left) and t=50 (right).

Also available:

### Some further notes

• If in the initial state the 16 monopoles are placed symmetric around the centre, the resulting evolution remains symmetric, as Interaction of 16 monopoles - symmetrical shows.
• The overall behaviour of the monopoles is the same if no-slip walls are used as boundary conditions, but the details are not: no-slip walls generated oppositely signed vorticity near the walls with which the monopoles interact when they reach the walls. This is explained on the page Interaction of 16 monopoles - not symmetrical, with no-slip walls.
• A single monopole in the centre of a domain simply rotates around its axis. If it is placed away from the centre, it feels its reflection in the stress-free nearest to it more than in the other walls. The result of this is that the monopole starts moving along an ellips-like path around the centre; the closer to the wall it starts, the faster it moves. See the page about the Bessel monopole.

The evolution of the vorticity distribution is computed with a Finite Difference Method which solves the two-dimensional vorticity (Navier-Stokes) equation. Time and distances are given in dimensionless units.

===> Some details on the computation presented on this page for those who are interested.