  In this work we design and analyse a Discrete de Rham (DDR) method for the
incompressible Navier-Stokes equations. Our focus is, more specifically, on the
SDDR variant, where a reduction in the number of unknowns is obtained using
serendipity techniques. The main features of the DDR approach are the support
of general meshes and arbitrary approximation orders. The method we develop is
based on the curl-curl formulation of the momentum equation and, through
compatibility with the Helmholtz-Hodge decomposition, delivers pressure-robust
error estimates for the velocity. It also enables non-standard boundary
conditions, such as imposing the value of the pressure on the boundary.
In-depth numerical validation on a complete panel of tests including general
polyhedral meshes is provided. The paper also contains an appendix where bounds
on DDR potential reconstructions and differential operators are proved in the
more general framework of Polytopal Exterior Calculus.
