Abstract The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a discretization of the Navier-Stokes equation that advects mass and momentum in a consistent manner. Incompressible flow with capillary forces is modeled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul d’Interface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to machine accuracy provided incompressibility is satisfied. To improve the stability of these methods momentum fluxes are advected in a manner “consistent” with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a 3mm-diameter falling raindrop and for an atomizing flow in air-water conditions.

中文翻译：

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用于交错网格上不可压缩流动的质量动量一致的流体体积方法

摘要 大密度对比流的计算是出了名的困难。为了减轻困难，我们考虑离散化 Navier-Stokes 方程，该方程以一致的方式对流质量和动量。对具有毛细管力的不可压缩流动进行建模，并在标记和单元类型的交错网格上执行离散化。流体体积法用于跟踪界面，高度函数法用于计算表面张力。体积分数的平流使用拉格朗日-显式/CIAM（Calcul d'Interface Affine par Morceaux）方法或Weymouth and Yao (WY) Eulerian-Implicit 方法进行。如果满足不可压缩性，WY 方法将流体质量守恒到机器精度。为了提高这些方法的稳定性，动量通量以与体积分数通量“一致”的方式平流，即动量的不连续性以与密度的不连续性相同的速度平流。为了找到以速度为中心的交错单元上的密度，执行了密度的辅助重建。该方法针对均匀流动中没有表面张力的液滴、在没有粘度或表面张力的非常大的密度比下在静止的载气中突然加速的液滴、针对开尔文-亥姆霍兹不稳定性、针对直径为 3 毫米的下落雨滴进行测试以及空气-水条件下的雾化流。为了找到以速度为中心的交错单元上的密度，执行了密度的辅助重建。该方法针对均匀流动中没有表面张力的液滴、在没有粘度或表面张力的非常大的密度比下在静止的载气中突然加速的液滴、针对开尔文-亥姆霍兹不稳定性、针对直径为 3 毫米的下落雨滴进行测试以及空气-水条件下的雾化流。为了找到速度集中的交错单元上的密度，执行密度的辅助重建。该方法针对均匀流动中没有表面张力的液滴、在没有粘度或表面张力的非常大的密度比下在静止的载气中突然加速的液滴、针对开尔文-亥姆霍兹不稳定性、针对直径为 3 毫米的下落雨滴进行测试以及空气-水条件下的雾化流。