  This paper introduces a formulation of the variable density incompressible
Navier-Stokes equations by modifying the nonlinear terms in a consistent way.
For Galerkin discretizations, the formulation leads to full discrete
conservation of mass, squared density, momentum, angular momentum and kinetic
energy without the divergence-free constraint being strongly enforced. In
addition to favorable conservation properties, the formulation is shown to make
the density field invariant to global shifts. The effect of viscous
regularizations on conservation properties is also investigated. Numerical
tests validate the theory developed in this work. The new formulation shows
superior performance compared to other formulations from the literature, both
in terms of accuracy for smooth problems and in terms of robustness.
