A (\(2N+1\))-equation model to simulate the flow of N (\(\textit{N}\ge 3\)) immiscible compressible fluids separated with interfaces is proposed. The model is based on the single velocity diffuse-interface method and includes \(N-1\) advection equations for fluid volume fractions. Solving the advection equations with a non-linear high-order scheme commonly results in the violation of the non-negativity constraint that any arbitrary partial sum of volume fractions should be in the interval [0, 1]. First, it is shown that this constraint can be met if the \(N-1\) advection equations are solved for some rational functions of volume fractions rather than for volume fractions themselves. The non-linear sub-cell slope reconstruction (MUSCL-type and THINC) with the proposed rational advection functions is proved to be non-oscillatory and provide the distribution of volume fractions satisfying the non-negativity constraint. Second, it is proved that the PV property (preservation of constant-pressure and constant-velocity equilibrium) is maintained providing that linear functions of volume fractions are used in the advection equations. We suggest two ways for resolving the contradiction in choosing the advection functions (functions of volume fractions) in accordance with the non-negativity constraint and the PV property. We also adopt two numerical methods—the Roe-type scheme and the HLLC scheme to solve the governing equations. Finally, the proposed numerical model is tested with several benchmark problems. The results obtained demonstrate robustness and effectiveness of the proposed numerical approach in solving multi-fluid flows with large interface deformations.

提出了模拟界面交界的N（\（\ textit {N} \ ge 3 \））不混溶可压缩流体流动的（\（2N + 1 \））方程模型。该模型基于单速度扩散界面方法，并且包含用于流体体积分数的\（N-1 \）对流方程。用非线性高阶方案求解对流方程通常会导致违反非负约束，即体积分数的任意部分和应在区间[0，1]中。首先，表明如果\（N-1 \）可以满足此约束平流方程是针对体积分数的某些有理函数而不是体积分数本身求解的。具有提出的有理对流函数的非线性子单元斜率重构（MUSCL型和THINC）被证明是非振荡性的，并提供了满足非负性约束的体积分数分布。其次，证明了在对流方程中使用体积分数的线性函数的情况下，可以保持PV特性（保持恒定压力和恒定速度的平衡）。我们提出了两种方法来解决根据非负约束和PV性质选择对流函数（体积分数函数）中的矛盾。我们还采用了两种数值方法-Roe型方案和HLLC方案来求解控制方程。最后，对提出的数值模型进行了几个基准测试。获得的结果证明了所提出的数值方法在解决具有大界面变形的多流体流方面的鲁棒性和有效性。