An Efficient Spectral-Projection Method for the Navier–Stokes Equations in Cylindrical Geometries: I. Axisymmetric Cases

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 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 terms of a fixed number of unknowns) and more efficient (in terms 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 nonaxisymmetric cases.