  The volume-filtering of the Navier-Stokes equations allows to consider the
effect that particles have on the fluid without further assumptions, but
closures arise of which the implications are not fully understood. 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 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 the
subfilter stress tensor requires modeling and should not be neglected. For
small filter widths, 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 and identify inconsistencies in previous studies of the
phase-averaged kinetic energy of the volume-filtered fluid velocity. Finally,
we propose a generally applicable form of the volume-filtered momentum equation
and its closures based on clear and well-founded assumptions and propose
guidelines for point-particle simulations based on the new findings.
