A new surface tension formulation for particle methods. (March 2020) - Duan, Riqiang; Sun, Chen; Jiang, Shengyao
Highlights: One measure in the new formulation is that the mean interface curvature is calculated using the surface divergence of the unit normal instead of the divergence used in the previous formulations. This measure completely removes errors stemming from interior particles due to numerical discretization. Another measure 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 the previous formulations. This treatment is motivated by the knowledge of the nature of the fractional dimensionality of the interface/curve, for which the macroscopic characteristics of the measured value are dependent on the measure scale. The new formulation is tested and verified using three numerical cases. It is shown that the new formulation is robust and accurate in calculating the surface tension, and meanwhile, it is able to adapt to different situations. 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.