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.

翻译：本文通过一致性地修改非线性项，提出了一种变密度不可压缩Navier-Stokes方程的新公式。对于Galerkin离散化，该公式能够在无需强加散度自由约束的情况下，实现质量、密度平方、动量、角动量以及动能的完全离散守恒。除了优异的守恒特性外，该公式还被证明能使密度场对全局平移保持不变性。本文还研究了粘性正则化对守恒特性的影响。数值试验验证了本工作中发展的理论。与文献中的其他公式相比，新公式无论是在光滑问题求解的精度方面，还是在鲁棒性方面，均表现出更优越的性能。