This work presents two unsplit geometric VOF schemes that extend the two-dimensional cellwise conservative unsplit (CCU) scheme [Comminal et al., J. Comput. Phys. 283 (2015) 582–608] to three dimensions. The novelty of the 3D-CCU schemes lies in the representation of the streaksurfaces of donating regions by polyhedral surfaces whose vertices are calculated with the 4th order Runge-Kutta scheme. Moreover, the advected liquid volumes are computed using a truncation algorithm [López et al., J. Comput. Phys. 392 (2019) 666–693] suited for arbitrary non-convex and self-intersecting polyhedra, which removes the need for tetrahedral decomposition. The 3D-CCU advection schemes were coupled to three interface reconstruction methods (Youngs' method, the Mixed Youngs-Centered scheme, and the Least-Square Fit algorithm). The resulting VOF methods were tested in classical benchmark advection tests, including translation, rigid-body rotation, shear and deformation flows. The proposed 3D-CCU schemes conserve the liquid volume and maintain the physical boundedness of liquid volume fractions to the machine precision. The 3D-CCU schemes perform favorably compared to other unsplit geometric VOF schemes when coupled to Youngs' interface reconstruction method. Moreover, the 3D-CCU schemes coupled to the Least-Square Fit algorithm are more accurate than most other VOF schemes that use a second-order accurate interface reconstruction, except those where a 3D extension of the Mosso-Swartz interface reconstruction is employed. The comparison of the different VOF schemes highlights the importance of coupling accurate interface reconstruction methods with accurate unsplit advection schemes.

这项工作提出了两个未拆分的几何 VOF 方案，它们扩展了二维单元方式保守未拆分 (CCU) 方案 [Comminal 等人，J. Comput. 物理。283 (2015) 582–608] 到三个维度。3D-CCU 方案的新颖之处在于用多面体表面表示捐赠区域的条纹表面，其顶点使用四阶 Runge-Kutta 方案计算。此外，使用截断算法计算平流液体体积 [López 等人，J. Comput. 物理。392 (2019) 666-693] 适用于任意非凸和自相交多面体，无需四面体分解。3D-CCU 对流方案与三种界面重建方法（Youngs 方法、混合 Youngs-Centered 方案和最小二乘拟合算法）相结合。由此产生的 VOF 方法在经典的基准平流测试中进行了测试，包括平移、刚体旋转、剪切和变形流。所提出的 3D-CCU 方案保存了液体体积，并将液体体积分数的物理有界性保持在机器精度上。与 Youngs 的界面重建方法耦合时，3D-CCU 方案与其他未分割的几何 VOF 方案相比表现良好。此外，与最小二乘拟合算法耦合的 3D-CCU 方案比大多数其他使用二阶精确界面重建的 VOF 方案更准确，除了采用 Mosso-Swartz 界面重建的 3D 扩展的那些方案。