Flow instabilities in the wake of a rounded square cylinder

Instabilities in the flow past a rounded square cylinder have been numerically studied in order to clarify the effects of rounding the sharp edges of a square cylinder on the primary and secondary instabilities associated with the flow. Rounding the edges was done by inscribing a quarter circle of radius $r$ in each edge of a square cylinder of height $d$. Nine cases of rounding were considered, ranging from a square cylinder ($r/d=0$) to a circular cylinder ($r/d=0.5$) with an increment of 0.0625. Each cross-section was numerically implemented in a Cartesian grid system by using an immersed boundary method. The key parameters are the Reynolds number (Re) and the edge-radius ratio ($r/d$). For low Re, the flow is steady and symmetric with respect to the centreline. Over the first critical Reynolds number ($Re_{c1}$), the flow undergoes a Hopf bifurcation to a time-periodic flow, termed the primary instability. As Re further increases, the onset of the secondary instability of three-dimensional (3D) nature is detected beyond the second critical Reynolds number ($Re_{c2}$). Rounding the sharp edges of a square cylinder significantly affects the flow topology, leading to noticeable changes in both instabilities. By employing the Stuart–Landau equation, we investigated the criticality of the primary instability depending upon $r/d$. The onset of the 3D secondary instability was detected by using Floquet stability analysis. The temporal and spatial characteristics of the dominant modes (A, B, QP) were described. The neutral stability curves of each mode were computed depending upon $r/d$.