Viscous evolution of 2D dipolar vortices

J.H.G.M. van Geffen and G.J.F. van Heijst
Fluid Dynamics Research 22, 191--213 (1998)

6. Concluding remarks

If a Lamb dipole is placed at the centre of a finite, rectangular domain with its axis parallel to the boundaries at y=± yw, then the dipole's velocity U is lower the narrower the domain is. If viscosity is present the vorticity of the dipole is gradually spread over a larger area, i.e. the radius a of the dipole increases as time evolves. At the same time the vorticity extrema decrease, as a result of which the dipole's velocity decreases.

The Lamb dipole initially has a linear relation between vorticity omega and streamfunction psi, namely omega=k²psi, where the value of k follows from ak=3.8317. As the dipole moves and grows, the omega,psi-relation becomes nonlinear, at first near the edge and the axis of the dipole, but as time goes on the nonlinearity spreads towards the vorticity extrema. As long as there is still a linear part in the omega,psi-relation around the extrema of vorticity, a k-value can be found. It appears that k decreases as function of time, but the product a*k increases with time. During this evolution the dipole retains its major characteristics: a more or less circular form and a more or less linear omega,psi-relation. These characteristics are named "Lamb-like".

As the dipole evolves further, a larger and larger part of the omega,psi-relation becomes nonlinear, until finally the entire omega,psi-relation can be described by omega=C*sinh(2*psi) for a certain constant C, which depends only on time. The dipole has then lost its main Lamb-like characteristics, although it is still circular to within a few per cent. The time scale at which this process takes place is determined mainly by the strength of the viscous effects; it does not so much depend on the strength of the initial dipole (a stronger initial dipole only moves faster).

Since the Lamb dipole appears to retain its characteristics for a long time, tests have been done with other 2D dipolar vorticity structures as initial vorticity distribution. The alternative dipoles studied have the same symmetry property as the Lamb dipole: two patches of oppositely signed vorticity which lie symmetrically about the line of motion. These dipoles evolve into a Lamb-like dipole, followed by a decay like that of the Lamb dipole itself. The time it takes to form a Lamb-like dipole depends on the initial vorticity distribution: the further it is away from a Lamb dipole, the more time it takes.

The main conclusion therefore is that a dipolar vortex with Lamb-like characteristics is a very stable vorticity structure in a viscous finite fluid, and initial vorticity distributions that are not too different from a Lamb dipole evolve to a dipolar structure with Lamb-like characteristics.

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