The immersed-boundary method is coupled to an incompressible-flow Reynolds-averaged Navier Stokes solver, based on a two-equation turbulence model, to perform unsteady numerical simulations of airflow past the NACA-0012 airfoil for several angles of attack and Reynolds numbers of 5.0×105 and 1.8×106. A preliminary study is performed to evaluate the sensitivity of the calculations to the computational mesh and to guide the creation of the computational cells for the unsteady calculations. Qualitative characterizations of the flow in the vicinity of the airfoil are obtained to assess the capability of locally refined grids to capture the thin boundary layers close to the airfoil leading edge as well as the wake flow emanating from the trailing edge. Quantitative analysis of aerodynamic force coefficients and wall pressure distributions are also reported and compared to experimental results and those from body-fitted grid simulations using the same solver to assess the accuracy and limitations of this approach. The immersed-boundary simulations compared well to the experimental and body-fitted results up to the occurrence of separation. After that point, neither computational approach provided satisfactory solutions.