A finite-volume code has been written to solve the complete, Reynolds-averaged Navier-Stokes equations around unconventional airfoils. The numerical algorithm is based on a flux-difference splitting form of a total variation diminishing (TVD) scheme. Various modifications to the scheme have been incorporated to provide a spatially second-order-accurate scheme in physical space. The scheme is conservative at steady state but employs nonconservative differencing during the integration to steady state to allow incorporation of implicit boundary conditions in the farfield. A zero-equation eddy viscosity model has been employed to represent the effects of turbulence. The code is validated by comparisons with flat plate and NACA 0012 data. Excellent results were obtained for both attached flow and shock induced separation cases. Numerical results are also presented for transonic flow over an unconventional airfoil and show good agreement.