  A semi-implicit fractional-step method that uses a staggered node layout and
radial basis function-finite differences (RBF-FD) to solve the incompressible
Navier-Stokes equations is developed. Polyharmonic splines (PHS) with
polynomial augmentation (PHS+poly) are used to construct the global
differentiation matrices. A systematic parameter study identifies a combination
of stencil size, PHS exponent, and polynomial degree that minimizes the
truncation error for a wave-like test function on scattered nodes. Classical
modified wavenumber analysis is extended to RBF-FDs on heterogeneous node
distributions and used to confirm that the accuracy of the selected 28-point
stencil is comparable to that of spectral-like, 6th-order Pad\'e-type finite
differences. The Navier-Stokes solver is demonstrated on two benchmark
problems, internal flow in a lid-driven cavity in the Reynolds number regime
$10^2\leq$Re$\leq10^4$, and open flow around a cylinder at Re=100 and 200. The
combination of grid staggering and careful parameter selection facilitates
accurate and stable simulations at significantly lower resolutions than
previously reported, using more compact RBF-FD stencils, without special
treatment near solid walls, and without the need for hyperviscosity or other
means of regularization.
