In this article developing incompressible viscous fluid flow in concentric and eccentric curved square annuli are numerically studied. A second order finite difference method based on the projection algorithm is implemented to solve the governing equations, including the full Navier–Stokes and continuity equations in a cylindrical coordinate system. To discretize the governing equations in the square annulus, a uniform staggered grid is used to enforce an exact second order numerical scheme. The effects of the governing nondimensional parameters involving the aspect ratio, curvature, Reynolds number, Dean number, and eccentricity on the flow field, both in developing and fully developed regions of the curved annular square duct, are studied in detail. The numerical results obtained indicate that the friction factor in the eccentric curved square annulus increases with the square root of the Dean number (κ^{1/2}) and the aspect ratio and decreases with the eccentricity. Furthermore, when the square root of the Dean number becomes larger than about 17.3, the friction factor increases linearly with the square root of the Dean number in the range of the current study.