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：本文通过一致地修改非线性项，提出了一种变密度不可压缩Navier-Stokes方程的公式化方法。对于伽辽金离散化，该公式在未强制施加无散度约束的条件下，实现了质量、密度平方、动量、角动量及动能的完全离散守恒。除优越的守恒特性外，该公式被证明可使密度场对整体平移保持不变。本文还研究了黏性正则化对守恒性质的影响。数值实验验证了本文理论的有效性。与文献中其他公式相比，新公式在处理光滑问题的精度和鲁棒性方面均展现出更优性能。