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This paper focuses on the characteristics of the regime C flow (Tatsuno & Bearman, J. Fluid Mech., vol. 211, 1990, pp. 157-182) around an oscillating circular cylinder in still water. The regime C flow is characterised by the formation of large-scale vortex cores arranged as opposed von Kármán vortex streets, resulting from a regular switching of vortex shedding directions with respect to the axis of oscillation. Both Floquet analysis and direct numerical simulations (DNS) are performed to investigate the two-(2-D) and three-dimensional (3-D) instabilities. The present study reveals that the low-wavenumber 3-D instability can emerge slightly before the 2-D instability in regime C. In total, five spanwise vortex modes were identified: (i) standing-wave pattern, S-mode; (ii) travelling-wave pattern, T-mode; (iii) mixed ST-mode; (iv) X-type vortex pattern, X-mode; and (v) U-type vortex pattern, U-mode. The modal analysis conducted in this study demonstrates that the vortex patterns and the corresponding spatial and temporal modulations of the dynamic loads of the S-, T- A nd mixed ST-modes are mainly induced by the 3-D instability of a single wavenumber. The characteristics of the X-mode are due to the superposition of the 3-D instabilities of multiple wavenumbers. The U-mode is dominated by a 2-D instability and its interaction with 3-D instabilities. The domain size dependence study demonstrates that the regime C flow is very sensitive to the spanwise length of the computational domain. The subcritical nature of the regime C flow is responsible for the discrepancy in the marginal stability curves obtained by independent Floquet stability analysis, DNS and physical experiments.