No abstract

The transition of the wake of a circular cylinder is investigated numerically via a stabilized finite element method for 150≤Re≤350. Both the flow and aerodynamic coefficients are studied. The onset of the three-dimensionality of the flow takes place via the mode-A instability at Re=200. At this Re, the flow exhibits pure mode-A type flow structures for t<1800. At larger times, the vortex dislocations appear spontaneously and destroy the spanwise periodicity in the flow. This confirms the hypothesis that the fully developed mode-A flow structures cannot exist without vortex dislocations. The appearance of dislocations leads to time variation in the vortex shedding frequency. They also lead to a reduction in the global aerodynamic parameters such as drag coefficient, rms value of lift coefficient, and dominant vortex shedding frequency. The vortex dislocations repetitively appear and disappear from the flow. The aerodynamic coefficients achieve a relatively lower value at the time instant when vortex dislocations appear in the flow. This leads to a low frequency modulation in the time variation of aerodynamic coefficients. The onset of mode-A is hysteretic. This is demonstrated in the present work via computations perhaps for the first time for increasing and decreasing Re. The transition from mode-A to mode-B vortex structures is gradual and not hysteretic. Mode-B is devoid of vortex dislocations and, therefore, the aerodynamic coefficients achieve a relatively larger value. The discontinuity in the variation of aerodynamic coefficients with Re is captured very well by the present computations. Unlike mode-A, the flow structures of the mode-B instability are restricted to the near wake.

Oblique shedding in the laminar regime for the flow past a nominally two-dimensional circular cylinder has been investigated numerically via a stabilized finite element method. No-slip condition on one of the sidewalls leads to the formation of a boundary layer which promotes oblique vortex shedding. Computations are carried out for three values of Reynolds number (Re): 60, 100, and 150. Cellular shedding is observed in all cases. Three cells are observed along the span for the Re=60 flow while only two cells are formed at Re=100 and 150. Spotlike vortex dislocations form at the junction of the cells. The frequency of the appearance of the dislocations increases with Re. Cellular shedding leads to low frequency modulation in the time histories of aerodynamic coefficients. Lowest value of drag is achieved at a time instant corresponding to the appearance of a new dislocation in the near wake. The vortex shedding frequency as well as the oblique angle of the primary vortices is found to vary with time for the Re=60 flow. Their variation is also related to the appearance of dislocations in the near wake. It is found that the vortex shedding frequency (Stθ) is related to the frequency observed for parallel shedding (St0) and the angle of the oblique vortices (θ) by the relation: Stθ=St0 cos θ. This relationship was proposed earlier for the case when the vortex shedding frequency and the oblique angle do not change with time. The velocity fluctuations are found to decrease with increase in θ. For the Re=100 and 150 flow, the oblique angle of the vortices and the shedding frequency outside the end cell do not change with time. However, θ and Stθ depend on the aspect ratio of the cylinder. The oblique shedding angle, for various lengths of endplate and Re, is found to vary linearly with the thickness of the boundary layer on the side wall.

Fluid–structure interaction (FSI) simulations are carried out to investigate vortex-induced vibrations of a sphere, mounted on elastic supports in all three spatial directions. The reduced velocity (${U}^{\ast } $) is systematically varied in the range ${U}^{\ast } = 4\text{{\ndash}} 9$, while the Reynolds number and reduced mass are held fixed at $\mathit{Re}= 300$ and ${m}^{\ast } = 2$, respectively. In the lock-in regime, two distinct branches are observed in the response curve, each corresponding to a distinct type of vortex shedding, namely, hairpin and spiral vortices. While shedding of hairpin vortices has been observed in several previous investigations of stationary and vibrating spheres, the shedding of intertwined, longitudinal spiral vortices in the wake of a vibrating sphere is reported herein for the first time. When the wake is in the hairpin shedding mode, the sphere moves along a linear path in the transverse plane, while when spiral vortices are shed, the sphere vibrates along a circular orbit. In the spiral mode branch, the simulations reveal hysteresis in the response amplitude at the beginning of the lock-in regime. Lower-amplitude vibrations are found as the sphere sheds hairpin vortices for increasing ${U}^{\ast } $ up until the beginning of the synchronization regime. On the other hand, higher-amplitude oscillations persist for the spiral mode as ${U}^{\ast } $ is decreased from the point of the start of the synchronization. The hairpin mode is found to be unstable for the value of reduced velocity where the spiral and hairpin solution branches merge together. When this point is approached along the hairpin solution branch, the sphere naturally transitions from shedding hairpin vortices and moving along a linear path to shedding spiral vortices and moving along a circular path in the transverse plane. The spiral mode was not observed in the work of Horowitz & Williamson (J. Fluid Mech., vol. 651, 2010, pp. 251–294), who studied experimentally the vibration modes of a freely rising or falling sphere and only reported zigzag vibrations. Our results suggest that this apparent discrepancy between experiments and simulations should be attributed to the fact that, for the range of governing parameters considered in the simulations, the elastic supports act to suppress streamwise vibrations, thus subjecting the sphere to a nearly axisymmetric elasticity constraint and enabling it to vibrate transversely along a circular path.