Title: An Efficient Spectral--Projection Method for the Navier-Stokes
       Equations in Cylindrical Geometries I. Axisymmetric cases}

Authors: J. M. Lopez and Jie Shen

Status: Submitted to J. Comput. Phys.

Abstract:

An efficient and accurate numerical scheme is presented for the
axisymmetric Navier-Stokes equations in primitive variables in a
cylinder. The scheme is based on a new spectral-Galerkin approximation
(\cite{Shen97}) for the space variables and a second-order projection
scheme for the time variable. The new spectral-projection scheme is
implemented to simulate the unsteady incompressible axisymmetric flow
with a singular boundary condition which is approximated to within a
desired accuracy by using a smooth boundary condition. A sensible
comparison is made with a standard second-order (in time and space)
finite difference scheme based on a stream function-vorticity
formulation and with available experimental data. The numerical
results indicate that both schemes  produce very reliable results and
that despite the singular boundary condition, the spectral-projection
scheme is still more accurate (in term of a fixed number of unknowns)
and more efficient (in term of CPU time required for resolving the
flow at a fixed Reynolds number to within a prescribed accuracy) than
the finite difference scheme. More importantly, the spectral-projection
scheme can be readily extended to  three-dimensional non-axisymmetric
cases.