Two-dimensional incompressible Navier-Stokes flow solver is developed using SIMPLER method to study the unsteady viscous flow physics over two-dimensional ellipses. Unsteady viscous flows over various thickness-to-chord ratios of 0.6, 0.8, 1.0, and 1.2 elliptic cylinders are simulated at different Reynolds numbers of 200, 400, and 1,000. This study is focused on the understanding the effects of Reynolds number and elliptic cylinder thickness on the drag and lift forces. The present numerical solutions are compared with available experimental and numerical results and show a good agreement. Through this study, it is observed that the Reynolds number and the cylinder thickness affect not only the frequency of the force oscillations but also the mean values and the amplitudes of the total drag and lift forces significantly.