C. H. K. Williamson

VORTEX FORMATION IN THE WAKE OF AN OSCILLATING CYLINDER

When a body oscillates laterally (cross-flow) in a free stream, it can synchronize the vortex formation frequency with the body motion frequency. This fundamental “lock-in” regions is but one in a whole series of synchronization regions, which have been found in the present paper, in an amplitude-wavelength plane (defining the body trajectory) up to amplitudes of five diameters. In the fundamental region, it is shown that the acceleration of the cylinder each half cycle induces the roll-up of the two shear layers close to the body, and thereby the formation of four regions of vorticity each cycle. Below a critical wavelength, each half cycle sees the coalescence of a pair of like-sign vortices and the development of a Karman-type wake. However, beyond this wavelength the like-sign vortices convect away from each other, and each of them pairs with an opposite-sign vortex. The resulting wake comprises a system of vortex pairs which can convect away from the wake centerline. The process of pairing causes the transition between these modes to be sudden, and this explains the sharp change in the character of the cylinder forces observed by Bishop and Hassan, and also the jump in the phase of the lift force relative to body displacement. At precisely the critical wavelength, only two regions of vorticity are formed, and the resulting shed vorticity is more concentrated than at other wavelengths. We interpret this particular case as a condition of “resonant synchronization”, and it corresponds with the peak in the body forces observed in Bishop and Hassans work.

Modes of vortex formation and frequency response of a freely vibrating cylinder

In this paper, we study the transverse vortex-induced vibrations of an elastically mounted rigid cylinder in a fluid flow. We use simultaneous force, displacement and vorticity measurements (using DPIV) for the first time in free vibrations. There exist two distinct types of response in such systems, depending on whether one has a high or low combined mass–damping parameter ( m *ζ). In the classical high-( m *ζ) case, an ‘initial’ and ‘lower’ amplitude branch are separated by a discontinuous mode transition, whereas in the case of low ( m *ζ), a further higher-amplitude ‘upper’ branch of response appears, and there exist two mode transitions. To understand the existence of more than one mode transition for low ( m *ζ), we employ two distinct formulations of the equation of motion, one of which uses the ‘total force’, while the other uses the ‘vortex force’, which is related only to the dynamics of vorticity. The first mode transition involves a jump in ‘vortex phase’ (between vortex force and displacement), ϕ vortex , at which point the frequency of oscillation ( f ) passes through the natural frequency of the system in the fluid, f ∼ f Nwater . This transition is associated with a jump between 2S [harr ] 2P vortex wake modes, and a corresponding switch in vortex shedding timing. Across the second mode transition, there is a jump in ‘total phase’, phis; total , at which point f ∼ f Nvacuum . In this case, there is no jump in ϕ vortex , since both branches are associated with the 2P mode, and there is therefore no switch in timing of shedding, contrary to previous assumptions. Interestingly, for the high-( m *ζ) case, the vibration frequency jumps across both f Nwater and f Nvacuum , corresponding to the simultaneous jumps in ϕ vortex and ϕ total . This causes a switch in the timing of shedding, coincident with the ‘total phase’ jump, in agreement with previous assumptions. For large mass ratios, m * = O (100), the vibration frequency for synchronization lies close to the natural frequency ( f * = f / f N ≈ 1.0), but as mass is reduced to m * = O (1), f * can reach remarkably large values. We deduce an expression for the frequency of the lower-branch vibration, as follows: formula here which agrees very well with a wide set of experimental data. This frequency equation uncovers the existence of a critical mass ratio , where the frequency f * becomes large: m * crit = 0.54. When m * m * crit , the lower branch can never be reached and it ceases to exist. The upper-branch large-amplitude vibrations persist for all velocities, no matter how high, and the frequency increases indefinitely with flow velocity. Experiments at m * m * crit show that the upper-branch vibrations continue to the limits (in flow speed) of our facility.

Three-dimensional wake transition

It is now well-known that the wake transition regime for a circular cylinder involves two modes of small-scale three-dimensional instability (modes “A” and “B”; Williamson, 1988), depending on the regime of Reynolds number (Re), although almost no understanding of the physical origins of these instabilities, or indeed their effects on near wake formation, have hitherto been made clear. There is now some strong interest in this problem, coming not only from experiment, but also from Direct Numerical Simulation, where, in some cases, these modes A and B have been found clearly (Thompson & Hourigan, 1996; Zhang et al., 1995; Henderson, 1995; Mittal & Balachandar, 1996). Much of the recent surge of activity concerning the wake transition and development of turbulence in wakes has been addressed comprehensively in a review paper. Williamson (1996a).

The existence of two stages in the transition to three‐dimensionality of a cylinder wake

The transition to three‐dimensionality in the near wake of a circular cylinder involves two successive transitions, each of which corresponds with a discontinuity in the Strouhal–Reynolds number relationship. The first discontinuity [between Reynolds numbers (Re) of 170 to 180] is associated with the inception of vortex loops, and it is hysteretic. The second discontinuity (between Re=230 to 260) corresponds with a change to a finer‐scale streamwise vortex structure. At this discontinuity there is no hysteresis, and it is suggested that two modes of vortex shedding alternate in time.

Defining a universal and continuous Strouhal–Reynolds number relationship for the laminar vortex shedding of a circular cylinder

The existence of a discontinuity in the Strouhal–Reynolds number relationship for the laminar vortex shedding of a cylinder is found to be caused by a change in the mode of oblique shedding. By ‘‘inducing’’ parallel shedding (from manipulating end conditions) the resulting Strouhal curve becomes completely continuous and agrees very well with the oblique‐shedding data, if it is transformed by S0=Sθ/cos θ (where Sθ is the Strouhal number corresponding with the oblique‐shedding angle θ). The curve also agrees with data from a completely different facility. This provides evidence that this Strouhal curve (S0) is universal (for a circular cylinder).

The effect of two degrees of freedom on vortex-induced vibration at low mass and damping

Although there are a great many papers dedicated to the problem of a cylinder vibrating transverse to a fluid flow (

Evolution of a single wake behind a pair of bluff bodies

Y

The instability of the shear layer separating from a bluff body

-motion), there are almost no papers on the more practical case of vortex-induced vibration in two degrees of freedom (

The natural and forced formation of spot-like ‘vortex dislocations’ in the transition of a wake

X,Y

Cooperative elliptic instability of a vortex pair

motion) where the mass and natural frequencies are precisely the same in both