Convergence and accuracy of kernel-based continuum surface tension models

Abstract

Numerical models for flows of immiscible fluids bounded by topologically complex interfaces possessing surface tension inevitably start with an Eulerian formulation. Here the interface is represented as a color function that abruptly varies from one constant value to another through the interface. This transition region, where the color function varies, is a thin O(h) band along the interface where surface tension forces are applied in continuum surface tension models. Although these models have been widely used since the introduction of the popular CSF method [BKZ92], properties such as absolute accuracy and uniform convergence are often not exhibited in interfacial flow simulations. These properties are necessary if surface tension-driven flows are to be reliably modeled, especially in three dimensions. Accuracy and convergence remain elusive because of difficulties in estimating first and second order spatial derivatives of color functions with abrupt transition regions. These derivatives are needed to approximate interface topology such as the unit normal and mean curvature. Modeling challenges are also presented when formulating the actual surface tension force and its local variation using numerical delta functions. In the following they introduce and incorporate kernels and convolution theory into continuum surface tension models. Here they convolve the discontinuous color function into a mollified function that can support accurate first and second order spatial derivatives. Design requirements for the convolution kernel and a new hybrid mix of convolution and discretization are discussed. The resulting improved estimates for interface topology, numerical delta functions, and surface force distribution are evidenced in an equilibrium static drop simulation where numerically-induced artificial parasitic currents are greatly mitigated