  In this paper, we present consistent and inconsistent discontinuous Galerkin
methods for incompressible Euler and Navier-Stokes equations with the kinematic
pressure, Bernoulli function and EMAC function. Semi- and fully discrete energy
stability of the proposed dG methods are proved in a unified fashion.
Conservation of total energy, linear and angular momentum is discussed with
both central and upwind fluxes. Numerical experiments are presented to
demonstrate our findings and compare our schemes with conventional schemes in
the literature in both unsteady and steady problems. Numerical results show
that global conservation of the physical quantities may not be enough to
demonstrate the performance of the schemes, and our schemes are competitive and
able to capture essential physical features in several benchmark problems.
