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：本文通过一致修改非线性项，提出了一种变密度不可压缩Navier-Stokes方程的表述格式。对于伽辽金离散，该格式在无需强加散度自由约束的情况下，实现了质量、密度平方、动量、角动量及动能的全离散守恒。除优越的守恒特性外，该格式还证明能使密度场对整体平移具有不变性。同时研究了粘性正则化对守恒性质的影响。数值实验验证了本文所发展的理论。与文献中其他格式相比，新格式在光滑问题的精度和鲁棒性方面均展现出更优性能。