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This paper introduces a formulation of the variable density incompressibleNavier-Stokes equations by modifying the nonlinear terms in a consistent way.For Galerkin discretizations, the formulation leads to full discreteconservation of mass, squared density, momentum, angular momentum and kineticenergy without the divergence-free constraint being strongly enforced. Inaddition to favorable conservation properties, the formulation is shown to makethe density field invariant to global shifts. The effect of viscousregularizations on conservation properties is also investigated. Numericaltests validate the theory developed in this work. The new formulation showssuperior performance compared to other formulations from the literature, bothin terms of accuracy for smooth problems and in terms of robustness.

A fully conservative and shift-invariant formulation for Galerkin discretizations of incompressible variable density flow