The classical vorticity-streamfunction formulation (VSF) can avoid the difficulty in the calculation of pressure gradient term of the Navier Stokes equation via eliminating pressure gradient term from the theoretical basis. Within this context we propose a general VSF, together with redefined vorticity and streamfunction, so as to realize numerically stable and reliable simulations of binary fluids with an arbitrary density contrast. By incorporating the interface-tracking phase-field model based on the conservative Allen-Cahn equation [Phys. Rev. E 94, 023311 (2016)], the binary flow simulation framework is established. Numerical tests are conducted using the Lattice Boltzmann method (LBM), which is usually regarded as an easy-to-use tool for solving the Navier–Stokes equation but generally suffers from the drawback of not being capable of enforcing incompressibility. The LBM herein functions as a numerical tool for solving the vorticity transport equation, the streamfunction equation, and the conservative Allen-Cahn equation. Three two-dimensional benchmark cases, i.e., the Capillary wave, the Rayleigh–Taylor instability, and the droplet splashing on a thin liquid film, are discussed in detail to verify the present methodology. Results show good agreements with both analytical predictions and literature data, as well as good numerical stability in terms of high density ratio and high Reynolds number. Overall, the general VSF inherits the intrinsic superiority of the classical VSF in enforcing incompressibility, and offers a useful and reliable alternative for binary flow modeling.

中文翻译：

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任意密度比不可压缩二元流的通用涡流函数公式

经典的涡流函数公式(VSF)通过从理论基础上消除压力梯度项，避免了纳维斯托克斯方程压力梯度项计算的困难。在此背景下，我们提出了通用的 VSF，以及重新定义的涡度和流函数，以便实现具有任意密度对比的二元流体的数值稳定且可靠的模拟。通过结合基于保守 Allen-Cahn 方程的界面跟踪相场模型 [Phys. Rev. E 94, 023311 (2016)]，建立了二元流模拟框架。数值测试是使用格子玻尔兹曼方法（LBM）进行的，该方法通常被认为是求解纳维-斯托克斯方程的易于使用的工具，但通常存在无法强制不可压缩性的缺点。本文中的 LBM 用作求解涡量传递方程、流函数方程和保守 Allen-Cahn 方程的数值工具。详细讨论了三个二维基准案例，即毛细管波、瑞利-泰勒不稳定性和薄液膜上的液滴飞溅，以验证当前的方法。结果显示与分析预测和文献数据均吻合良好，并且在高密度比和高雷诺数方面具有良好的数值稳定性。总体而言，通用 VSF 继承了经典 VSF 在强制不可压缩性方面的内在优势，并为二元流建模提供了有用且可靠的替代方案。