  We present a novel framework to deal with static and moving immersed
boundaries (IB). In this strategy, called Volume-Filtering Immersed Boundary
(VFIB) method, transport equations are derived by filtering the Navier-Stokes
equations and accounting for stresses at the solid-fluid interface. The result
is that boundary conditions that normally apply on the interface are
transformed into bodyforces that apply on the right-hand side of the filtered
transport equations. The filter width acts as a parameter that controls the
level of resolution. The IB is considered well-resolved if the filter width is
much smaller than the characteristic corrugation scale of the interface. There
are several innovations in this IB method. First, it sheds light on the role of
the internal flow which arises when the transport equations are solved inside
the IB. We show that, it is essential to separate stresses due to the external
and internal fluids in order to get accurate forces and provide a method to do
so. Second, we show that volumes associated with Lagrangian forcing points on
the boundary depend on the local topology of the surface. We provide a method
to compute these volumes using a triangle tessellation of the interface and the
surface density function. Third, we provide an efficient procedure to compute
the solid volume fraction, thus, enabling tagging interior/exterior cells. This
volume fraction is also involved in the procedure to separate stresses due to
the external fluid from the total stresses. Fourth, we show a path forward to
extend the VFIB method to Large Eddy Simulations involving IBs. Lastly, we
apply the VFIB in several numerical tests involving two- and three- dimensional
static and moving IBs. We show greatly improved results compared to prior IB
methods. Further, we test several filter kernels and show that, for
well-resolved IBs, the choice of the kernel plays little role.
