A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates

A finite-difference scheme for the direct simulation of the incom pressible time-dependent three-dimensional Navier-Stokes equations in cylindrica I coordinates is presented. The equations in primitive variables(v(r), v(theta), v(z) and p) are solved by a fractional-step method together with an approximate-factorization technique. Cylindrical coordinates are singular at the axis; the introduction of the radial flux q(r) = r . v(r) on a staggered grid simplifies the treatment of the region at r = 0. The method is tested by comparing the evolution of a free vortex ring and its collision with a wall with the theory, experiments, and other numerical results. The formation of a tripolar vortex, where the highest vorticity is at r = 0, is also considered. Finally to emphasize the accurate treatment near the axis, the motion of a Lamb dipole crossing the origin is simulated. (C) 1996 Academic Press. Inc.