  In this paper, we propose a unified numerical framework for the
time-dependent incompressible Navier--Stokes equation which yields the $H^1$-,
$H(\text{div})$-conforming, and discontinuous Galerkin methods with the use of
different viscous stress tensors and penalty terms for pressure robustness.
Under minimum assumption on Galerkin spaces, the semi- and fully-discrete
stability is proved when a family of implicit Runge--Kutta methods are used for
time discretization. Furthermore, we present a unified discussion on the
penalty term. Numerical experiments are presented to compare our schemes with
classical schemes in the literature in both unsteady and steady situations. It
turns out that our scheme is competitive when applied to well-known benchmark
problems such as Taylor--Green vortex, Kovasznay flow, potential flow, lid
driven cavity flow, and the flow around a cylinder.
