A new surface tension formulation in smoothed particle hydrodynamics for free-surface flows

Abstract

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.