  Immersed methods discretize boundary conditions for complex geometries on
background Cartesian grids. This makes such methods especially suitable for
two-way coupled flow-body problems, where the body mechanics are partially
driven by hydrodynamic forces. However, for the vorticity-velocity form of the
Navier-Stokes equations, existing immersed geometry discretizations for two-way
coupled problems only achieve first order spatial accuracy near solid
boundaries. Here we introduce a sharp-interface approach based on the immersed
interface method to handle the one- and two-way coupling between an
incompressible flow and one or more rigid bodies using the 2D
vorticity-velocity Navier-Stokes equations. Our main contributions are
three-fold. First, we develop and analyze a moving boundary treatment for sharp
immersed methods that can be applied to PDEs with implicitly defined boundary
conditions, such as those commonly imposed on the vorticity field. Second, we
develop a two-way coupling methodology for the vorticity-velocity Navier-Stokes
equations based on control-volume momentum balance that does not require the
pressure field. Third, we show through extensive testing and validation that
our resulting flow-body solver reaches second-order accuracy for most practical
scenarios, and provides significant efficiency benefits compared to a
representative first-order approach.
