Study and derivation of closures in the volume-filtered framework for particle-laden flows

The volume filtering of the governing equations of a particle-laden flow allows for simulating the fluid phase as a continuum, and accounting for the momentum exchange between the fluid and the particles by adding a source term in the fluid momentum equations. The volume filtering of the Navier-Stokes equations allows consideration of the effect that particles have on the fluid without further assumptions, but closures arise of which the implications are not fully understood. It is common to either neglect these closures or model them using assumptions of which the implications are unclear. In the present paper, we carefully study every closure in the volume-filtered fluid momentum equation and investigate their impact on the momentum and energy transfer dependent on the filtering characteristics. We provide an analytical expression for the viscous closure that arises because the filter and spatial derivative in the viscous term do not commute. An analytical expression for the regularization of the particle momentum source of a single sphere in the Stokes regime is derived. Furthermore, we propose a model for the subfilter stress tensor, which originates from filtering the advective term. The model for the subfilter stress tensor is shown to agree well with the subfilter stress tensor for small filter widths relative to the size of the particle. We show that, in contrast to common practice, the subfilter stress tensor requires modelling and should not be neglected. For filter widths comparable to the particle size, we find that the commonly applied Gaussian regularization of the particle momentum source is a poor approximation of the spatial distribution of the particle momentum source, but for larger filter widths, the spatial distribution approaches a Gaussian. Furthermore, we propose a modified advective term in the volume-filtered momentum equation that consistently circumvents the common stability issues observed at locally small fluid volume fractions. Finally, we propose a generally applicable form of the volume-filtered momentum equation and its closures based on clear and well-founded assumptions. Based on the new findings, guidelines for point-particle simulations and the filter width with respect to the particle size and fluid mesh spacing are proposed.