When the surface tension force acts on the multi-layer particles near the free surface, the integral of the previous surface delta function through the free surface is less than 1 because of the boundary truncation and kernel truncation, which leads to the problem that the numerical results cannot converge to the theoretical solution. Although a few studies have recognized this problem, it has not been completely solved. In the present work, a new formulation of the continuum surface force (CSF) model in smoothed particle hydrodynamics (SPH) is developed for free-surface flows, in which two measures are taken to correct the shortcomings of the commonly used formulations. In the first measure, a new surface delta function that meets the normalization requirement is proposed to solve the problem that the integral of the previous surface delta function through the free surface is less than 1 when the surface tension force acts on the multi-layer particles near the free surface. This normalization measure is key to make the numerical results with surface tension converge to the theoretical solution. In the second measure, the normal vector and curvature of the free surface are derived by the finite particle method (FPM) to improve their calculation accuracy. Through the one-dimensional tests, the new surface delta function is analyzed and compared with existing formulations. Through the two-dimensional tests on a unit circle, the new normal vector and curvature of the particles near the free surface are implemented and also compared with previous formulations. Finally, the proposed formulation is validated using eight two-dimensional and three-dimensional numerical cases with surface tension. The results show that the proposed formulation solves the problem that the numerical results of the previous formulation cannot converge to the theoretical solution, and it is accurate and robust in calculating the free-surface flows with surface tension.

当表面张力作用在自由表面附近的多层颗粒上时，由于边界截断和核截断，通过自由表面的先前表面δ函数的积分小于1，这导致了数值上的问题。结果不能收敛到理论解。尽管一些研究已经意识到了这个问题，但尚未完全解决。在目前的工作中，开发了一种用于自由表面流动的光滑表面流体动力学（SPH）中连续表面力（CSF）模型的新公式，其中采取了两种措施来纠正常用公​​式的缺点。首先，提出了一种满足归一化要求的新的表面δ函数，以解决当表面张力作用于自由表面附近的多层粒子时，通过自由表面的先前表面δ函数的积分小于1的问题。该归一化措施对于使具有表面张力的数值结果收敛到理论解是关键。在第二种方法中，通过有限粒子方法（FPM）导出自由表面的法向矢量和曲率，以提高其计算精度。通过一维测试，可以分析新的表面增量功能，并将其与现有配方进行比较。通过对单位圆的二维检验，新的法向矢量和自由曲面附近粒子的曲率得以实现，并与以前的公式进行了比较。最后，使用八个具有表面张力的二维和三维数值案例验证了所提出的公式。结果表明，所提出的公式解决了先前公式的数值结果不能收敛于理论解的问题，并且在计算具有表面张力的自由表面流时是准确而稳健的。