| The treatment of multiscale surface phenomena is a fundamental problem in the computation of spray atomization, breaking waves, and many other multiphase flows. Existing methods typically assume that the flow structures are either fully resolved on the computational grid, or are entirely unresolved (i.e., point particles). However, real droplet-producing flows often involve a continuous cascade of scales from large initial structures that break up into smaller and smaller structures and eventually into droplets, and neither method is applicable for the intermediate scales. This work proposes an alternate formulation which is applicable for flows over the entire range of scales from resolved to unresolved.; We define the mass and momentum equations for each fluid in the system using a formulation that is well-defined over the entire computational domain, including outside the region where the fluid is located. This formulation is then spatially filtered, as in the derivation of large-eddy simulations for turbulent flows but with a filter that is independent of the locations of the fluid interfaces. The result is a flowfield in which the flow velocities and the surface structures have been separated into resolved and unresolved components, and a set of equations for the evolution of the resolved components.; This formulation can then be combined with models for the specific flow under consideration, in order to develop a complete simulation method. These models include models for the viscous stress tensor, for the interactions between fluids at the phase boundaries, and for the effects of the unresolved-scale flow on the spatially-averaged momentum advection. With appropriate choices of models, many existing methods such as immersed boundary methods, continuuum-surface-force representations of surface tension, interpenetrating-continua formulations, and point-particle methods can be recovered as special cases of this formulation. As a result, the formulation can be used to evaluate the range of applicability of these methods and to suggest enhancements to them.; Two demonstrations of this formulation are shown. Computations of an oscillating viscous droplet are used to demonstrate the fully-resolved case where the unresolved-scale models can be neglected. These computations also demonstrate a method for tracking the surface motion which is suggested by the formulation. Computations of flow around circular and spherical particles are then used to demonstrate the behavior of the method for partly-resolved and unresolved flows. These calculations illustrate how the formulation can be used to evaluate existing methods for momentum exchange---in this case, point-particle methods---and to suggest enhancements for these methods. In particular, we find that the applicability of the point-particle assumption is strongly dependent on the Reynolds number, and that for cases where a significant wake is present, point-force models can result in inaccurate velocity fields even when the particle is nearly two orders of magnitude smaller than the filter. Further, we find that point-particle models can be enhanced significantly by including an axial dipole term to represent the unresolved-scale momentum advection effects in the near-particle flow. |
