  We consider a numerical approach for the incompressible surface Navier-Stokes
equation. The approach is based on the covariant form and uses discrete
exterior calculus (DEC) in space and a semi-implicit discretization in time.
The discretization is described in detail and related to finite difference
schemes on staggered grids in flat space for which we demonstrate second order
convergence. We compare computational results with a vorticity-stream function
approach for surfaces with genus 0 and demonstrate the interplay between
topology, geometry and flow properties. Our discretization also allows to
handle harmonic vector fields, which we demonstrate on a torus.
