Development of a Spatially-Filtered Method for Calculation of Multiphase Flow

Computer simulations of multiphase flows are currently limited to two distinct regimes: Fluid elements that are much larger than the computational grid cells can be modeled using fully-resolved calculations, and fluid elements that are much smaller than the grid cells can be modeled using point-particle approximations. However, many flows involve fluid elements for which neither of these limits is appropriate. For simulations of spray atomization, in particular, the grid size needs to be small enough to resolve the details of the injected fluid stream, yet resolving the smallest droplets produced would require an impossibly large number of grid points. To simulate spray atomization or other flows with a wide range of fluid element sizes, then, it is necessary to develop a computational framework that can bridge the fully-resolved and point-particle regimes. The development of such a framework is the topic of this paper. We begin by borrowing the mathematical framework of “large-eddy simulations” of turbulence—the equations for the flowfield are mathematically filtered, producing a set of equations which are smooth above a certain size and contain a set of terms which model the effects of the smaller scales. Because the filtering provides a mathematically valid smoothing of the fluid interfaces, it is also a natural way of simulating fully-resolved fluid elements on a fixed grid. The subfilter-scale terms can be shown to be zero in the fully-resolved limit, and so fully-resolved simulations provide an appropriate test of the basic filtered-multiphase method, independent of the complex subgrid modeling that would be necessary for unresolved elements. In anticpation of future calculations of droplets, we demonstrate the method with two-dimensional calculations of fluid cylinder oscillations, and compare the results to an analytical solution.