The evolution of single elliptic vortex rings for initial aspect ratio (AR) =2,4,6 has been studied. The incompressible Navier-Stokes equations are solved by a dealiased pseudo-spectral method with 643 grid points in a periodic cube. We find that there are three kinds of vortex motion as AR increases and bifurcation occurs at certain AR. The processes of advection, interaction and decay of vortex ring axe discussed. Numerical results coincide with experiments and other authors' numerical simulation.

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Fourier expansions in the radial direction for unbounded ows expressed in a cylindrical coordinate system are proposed. By appropriate coordinate mapping and periodic extension in the r direction, periodic boundary conditions required by Fourier expansions and in nite di erentiability demanded by spectral convergence are established. Appropriate zero factors for the general Fourier expansions are given at the axis and at in nity in order to remove the numerical singularity at r = 0 and to satisfy all the boundary conditions. The e ectiveness of these expansions are demonstrated by the simulation of steady axisymmetrical vortex rings in ideal uid and the numerical simulation of the head-on collision of two coaxial, equal and opposite vortex rings.