Global stability of the flow past a stepped cylinder

Abstract

We investigate the global instability mechanism of the flow past a three-dimensional stepped cylinder. A comprehensive study is performed for different diameter ratios of the two joined cylinders ( $r=D/d$ ) ranging from $r=1.1$ to $r=4$ . Independently of $r$ , the spectrum of the linearised Navier–Stokes operator reveals a pair of complex conjugate eigenvalues, with Strouhal number $St \approx 0.11$ . The initial transition is triggered by a two-dimensional mechanism of the larger cylinder only, not affected by the presence of the junction and the smaller cylinder ( $Re_{D,cr}\approx 47$ ). The structural sensitivity analysis is used to identify where the instability mechanism acts. The onset of transition is solely localised in the large cylinder wake (L cell), where the wavemaker has two symmetric lobes across the separation bubble. When the Reynolds number increases, a second and a third unstable pair of complex conjugate eigenvalues appears. They are localised in the small cylinder (S) wake and modulation (N) region. For any $r$ , the appearance of unstable eigenmodes resembling the three cells S–N–L in the wake is observed. The nonlinear simulation results support this finding, in contrast with the previous classification of the laminar vortex shedding in direct (L–S) and indirect (L–N–S) modes interaction Lewis & Gharib (Phys. Fluids, vol. 4, 1992, pp. 104–117). This result indicates that each cell undergoes a supercritical Hopf bifurcation for any $r$ . As $r$ approaches $1$ , the modal linear stability results also show an unstable eigenmode in the wake of the small cylinder resembling a new modulation cell, named N2, similar to the N cell but mirrored with respect to the junction plane.