In this study we have developed a flexible and efficient numerical scheme for the simulation of three–dimensional incompressible flows in spherical coordinates. The main idea, inspired by a similar strategy as  for cylindrical coordinates, consists of a change of variables combined with a discretization on a staggered mesh and the special treatment of few discrete terms that remove the singularities of the Navier–Stokes equations at the sphere centre and along the polar axis. This new method alleviates also the time step restrictions introduced by the discretization around the polar axis while it still suffers from strong limitations if convection at the sphere centre dominates the flow. The scheme is second–order accurate in space and is verified and validated by computing numerical examples that are compared with similar results produced by other codes or available from the literature. The method can cope with flows evolving in the whole sphere, in a spherical shell and in a sector without any change and, thanks to the flexibility of finite–differences, it can employ generic mesh stretching (in two of the three directions) and complex boundary conditions.
- Singularity at the centre
- Singularity at the polar axis
- Spherical coordinates