A Spectral Method for Unbounded Flow in a Cylindrical Coordinate System

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 innnite diierentiability demanded by spectral convergence are established. Appropriate zero factors for the general Fourier expansions are given at the axis and at innnity in order to remove the numerical singularity at r = 0 and to satisfy all the boundary conditions. The eeectiveness 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.