Linear stability of the steady flow past rectangular cylinders

Abstract

The primary instability of the flow past rectangular cylinders is studied to comprehensively describe the influence of the aspect ratio$AR$and of rounding the leading- and/or trailing-edge corners. Aspect ratios ranging between$0.25$and$30$are considered. We show that the critical Reynolds number ($\textit {Re}_c$) corresponding to the primary instability increases with the aspect ratio, starting from$\textit {Re}_c \approx 34.8$for$AR=0.25$to a value of$\textit {Re}_c \approx 140$for$AR = 30$. The unstable mode and its dependence on the aspect ratio are described. We find that positioning a small circular cylinder in the flow modifies the instability in a way strongly depending on the aspect ratio. The rounded corners affect the primary instability in a way that depends on both the aspect ratio and the curvature radius. For small$AR$, rounding the leading-edge corners has always a stabilising effect, whereas rounding the trailing-edge corners is destabilising, although for large curvature radii only. For intermediate$AR$, instead, rounding the leading-edge corners has a stabilising effect limited to small curvature radii only, while for$AR \geqslant 5$it has always a destabilising effect. In contrast, for$AR \geqslant 2$rounding the trailing-edge corners consistently increases$\textit {Re}_c$. Interestingly, when all the corners are rounded, the flow becomes more stable, at all aspect ratios. An explanation for the stabilising and destabilising effect of the rounded corners is provided.