This study attempts to compare the behavior of concave and convex square cylinders in laminar incompressible flow to investigate the effect of geometry on primary instability. Cases are simulated in finite volume-based software OpenFOAM and developed and unsteady flow characteristics are compared at Reynolds number (Re hereafter) 55. Stuart–Landau equation was used to calculate the critical Re, which is the onset of primary instability. It was found that the critical Re for the concave cylinder, Re = 53.76, is higher than that for the convex cylinder, Re = 43.12, and the value of the coefficient of drag came out to be 1.41 and 1.612 for the concave and convex cylinder, respectively. This difference was attributed to the stronger shear layer for convex cylinder than concave, which was responsible for its higher coefficient of drag. Additionally, velocity and vorticity plots were analyzed for both cylinders to understand the flow field better.