  In this paper, we design and analyze staggered discontinuous Galerkin methods
of arbitrary polynomial orders for the stationary Navier-Stokes equations on
polygonal meshes. The exact divergence-free condition for the velocity is
satisfied without any postprocessing. The resulting method is pressure-robust
so that the pressure approximation does not influence the velocity
approximation. A new nonlinear convective term that earning non-negativity is
proposed. The optimal convergence estimates for all the variables in $L^2$ norm
are proved. Also, assuming that the rotational part of the forcing term is
small enough, we are able to prove that the velocity error is independent of
the Reynolds number and of the pressure. Furthermore, superconvergence can be
achieved for velocity under a suitable projection. Numerical experiments are
provided to validate the theoretical findings and demonstrate the performances
of the proposed method.
