Abstract In this paper, a new formulation based on the continuum surface force model (CSF) is developed for particle methods. In the new formulation, two measures are taken to amend the two drawbacks faced by previous formulations of the CSF model. One measure is that the interface curvature is calculated using the surface divergence of the unit normal instead of the divergence as done in previous formulations. This measure completely removes errors artificially stemming from interior particles that are caused by numerical discretization. Another measure taken is that the size of the particle interaction domain is determined based on the physical characteristic length, rather than taken as a fixed multiple of the particle spacing, as in previous formulations. This treatment is based on the knowledge of the nature of the fractional dimensionality of the surface/curve, for which the measured values of macroscopic characteristics are dependent on the measurement scale. Through the curvature calculation of a unit circle, the relation between the size of the particle interaction domain and the critical size of droplet/bubble breakup is established. The former, which is taken as approximately 0.38 times the latter, is accurate enough to obtain a reasonable interface curvature, even with severe irregularity. The new formulation is tested and verified using three numerical cases, including the oscillation and decay of a square drop under surface tension, the coalescence of two drops after a head-on collision, and the Rayleigh-Taylor instability. It is shown that the new formulation is robust and accurate in calculating the surface tension and is able to adapt to different situations.

中文翻译：

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粒子方法的新表面张力公式

摘要 在本文中，为粒子方法开发了一种基于连续表面力模型 (CSF) 的新公式。在新的公式中，采取了两项措施来修正以前的 CSF 模型公式所面临的两个缺点。一种措施是使用单位法线的表面发散而不是以前公式中所做的发散来计算界面曲率。该措施完全消除了由数值离散化引起的内部粒子人为产生的误差。采取的另一个措施是，粒子相互作用域的大小是根据物理特征长度确定的，而不是像以前的公式一样，将其视为粒子间距的固定倍数。这种处理基于对表面/曲线的分数维数性质的了解，为此，宏观特征的测量值取决于测量尺度。通过单位圆的曲率计算，建立了粒子相互作用域尺寸与液滴/气泡破裂临界尺寸之间的关系。前者大约是后者的 0.38 倍，即使存在严重的不规则性，也足以获得合理的界面曲率。新公式使用三个数值案例进行测试和验证，包括表面张力下方形液滴的振荡和衰减、正面碰撞后两滴的合并以及瑞利-泰勒不稳定性。