  We solve the incompressible Navier-Stokes equations using a lattice Green's
function (LGF) approach, including immersed boundaries (IB) and adaptive mesh
refinement (AMR), for external flows with one homogeneous direction (e.g.
infinite cylinders of arbitrary cross-section). We hybridize a Fourier
collocation (pseudo-spectral) method for the homogeneous direction with a
specially designed, staggered-grid finite-volume scheme on an AMR grid. The
Fourier series is also truncated variably according to the refinement level in
the other directions. We derive new algorithms to tabulate the LGF of the
screened Poisson operator and viscous integrating factor. After adapting other
algorithmic details from the fully inhomogeneous case, we validate and
demonstrate the new method with transitional and turbulent flows over a
circular cylinder at $Re=300$ and $Re=12,000$, respectively.
