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：本文通过一致的方式修改非线性项，提出了变密度不可压缩纳维-斯托克斯方程的一种新公式。对于伽辽金离散，该公式在无需强约束散度自由条件的情况下，实现了质量、密度平方、动量、角动量和动能的完全离散守恒。除了有利的守恒性质外，该公式还使密度场具有全局平移不变性。本文还研究了粘性正则化对守恒性质的影响。数值验证支持了本文发展的理论。与文献中的其他公式相比，新公式在光滑问题的精度和鲁棒性方面均表现出更优的性能。