In this study, a higher order compact (HOC) finite difference scheme is proposed for the simulation of multiphase flows with surface tension effects by coupling the pure streamfunction formulation of Navier–Stokes (N–S) equations with the level set method. Unlike most of the existing methods for multiphase flows that solve the incompressible Navier–Stokes equations in the velocity–pressure form or the stream-function vorticity form, we recast the governing equations into the biharmonic form of the N–S equations which carries stream-function $\psi$ as the only variable to be computed. The coupling of this formulation with level set (LS) enables the simulation of two phase flows by the core computation of only the single variable $\psi$ free from discontinuities in the whole physical domain. In the process, we also propose a novel hybrid HOC algorithm for post-processing the pressure on either side of the interface including the jump across it. The robustness of this method is established through a series of benchmark problems such as dispersion of capillary waves, static bubble and oscillating bubble at equilibrium in a fluid at rest, and deforming circular drops in shear flows. We compare our numerical results with available benchmark results and excellent match is obtained in all the cases along with accurate representation of overall trends. Our approach is proved to be very efficient in terms of its easy implementation, relatively low computational time and high order of accuracy.

在这项研究中，通过将Navier–Stokes（NS）方程的纯流函数公式与水平集方法相结合，提出了一种高阶紧凑（HOC）有限差分方案来模拟具有表面张力效应的多相流。与大多数现有的多相流方法可以解决速度-压力形式或流函数涡度形式的不可压缩的Navier-Stokes方程不同，我们将控制方程改写为N-S方程的双调和形式，该方程携带流。功能$\psi$作为唯一要计算的变量。该公式与水平集（LS）的耦合使得仅通过单个变量的核心计算就可以模拟两个相流$\psi$在整个物理域中都没有间断。在此过程中，我们还提出了一种新颖的混合HOC算法，用于对接口任一侧的压力进行后处理，包括跨越接口的跳跃。该方法的鲁棒性是通过一系列基准问题建立的，例如毛细波的分散，静态气泡和静止流体中处于平衡状态的振荡气泡以及剪切流中的圆形液滴变形。我们将数值结果与可用的基准结果进行比较，并在所有情况下均获得了出色的匹配，并精确地表示了总体趋势。我们的方法在易于实现，相对较低的计算时间和较高的准确性方面被证明是非常有效的。