Linear stability of the steady flow past rectangular cylinders

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 (Re-c) corresponding to the primary instability increases with the aspect ratio, starting from Re-c approximate to 34.8 for AR = 0.25 to a value of Re-c approximate to 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 >= 5 it has always a destabilising effect. In contrast, for AR >= 2 rounding the trailing-edge corners consistently increases 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.