Presented article studies the secondary wake mode instability analysis behind a two-dimensional cylinder using OpenFOAM®. The instability of fluid flow is examined by obtaining the disturbance fields around the stable base flow solution. The objective of presented paper is to modify the existing icoFoam solver in OpenFOAM® to solve the linearized Navier–Stokes equation. The novelty of work lies in the utilization of an open source software—OpenFOAM® to predict the secondary wake (SW) instability around the base flow over two-dimensional circular cylinder. Modification of an inbuilt solver is done by altering the standard incompressible solver, i.e., icoFoam and its associated libraries to create a new modified solver named “stabilityFoam” (Code source link: https://github.com/ravikgpiit/stabilityFoam . The numerical simulation is investigated near secondary wake (SW) mode instability (Re \(\sim \) 110.8) (Verma and Mittal in Phys Fluids 23(12):121701, [36]). In this regard, two Reynolds numbers are chosen as 100 (sub-critical) and 150 (supercritical). The stable base flow is simulated prior to the modified solver. We found that the flow shows stable transient energy growth characteristics for sub-critical Reynolds Numbers 100, while it is unstable and perturbations grow boundlessly at supercritical Reynolds number 150. The motivation of this study is to demarcate the secondary wake instability in flow over cylinder at sub-critical and supercritical Reynolds numbers utilizing OpenFOAM®.