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▪ Abstract This review summarizes fundamental results and discoveries concerning vortex-induced vibration (VIV), that have been made over the last two decades, many of which are related to the push to explore very low mass and damping, and to new computational and experimental techniques that were hitherto not available. We bring together new concepts and phenomena generic to VIV systems, and pay special attention to the vortex dynamics and energy transfer that give rise to modes of vibration, the importance of mass and damping, the concept of a critical mass, the relationship between force and vorticity, and the concept of “effective elasticity,” among other points. We present new vortex wake modes, generally in the framework of a map of vortex modes compiled from forced vibration studies, some of which cause free vibration. Some discussion focuses on topics of current debate, such as the decomposition of force, the relevance of the paradigm flow of an elastically mounted cylinder to more complex systems, and the relationship between forced and free vibration.

Since the review of periodic flow phenomena by Berger & Wille (1972) in this journal, over twenty years ago, there has been a surge of activity regarding bluff body wakes. Many of the questions regarding wake vortex dynamics from the earlier review have now been answered in the literature, and perhaps an essential key to our new understandings (and indeed to new questions) has been the recent focus, over the past eight years, on the three-dimensional aspects of nominally two-dimensional wake flows. New techniques in experiment, using laser-induced fluorescence and PIV (Particle-Image-Velocimetry), are vigorously being applied to wakes, but interestingly, several of the new discoveries have come from careful use of classical methods. There is no question that strides forward in understanding of the wake problem are being made possible by ongoing threedimensional direct numerical simulations, as well as by the surprisingly successful use of analytical modeling in these flows, and by secondary stability analyses. These new developments, and the discoveries of several new phenomena in wakes, are presented in this review.

These experiments, involving the transverse oscillations of an elastically mounted rigid cylinder at very low mass and damping, have shown that there exist two distinct types of response in such systems, depending on whether one has a low combined mass-damping parameter (low m* ), or a high mass-damping (high m* ). For our low m* , we "nd three modes of response, which are denoted as an initial amplitude branch, an upper branch and a lower branch. For the classical Feng-type response, at high m* , there exist only two response branches, namely the initial and lower branches. The peak amplitude of these vibrating systems is principally dependent on the mass-damping (m* ), whereas the regime of synchronization (measured by the range of velocity ;*) is dependent primarily on the mass ratio, m*. At low (m* ), the transition between initial and upper response branches involves a hysteresis, which contrasts with the intermittent switching of modes found, using the Hilbert transform, for the transition between upper}lower branches. A 1803 jump in phase angle is found only when the #ow jumps between the upper}lower branches of response. The good collapse of peak-amplitude data, over a wide range of mass ratios (m*"1}20), when plotted against (m*#C ) in the &&Gri$n'' plot, demonstrates that the use of a combined parameter is valid down to at least (m*#C ) &0)006. This is two orders of magnitude below the &&limit'' that had previously been stipulated in the literature, (m*#C ) '0)4. Using the actual oscillating frequency ( f ) rather than the still-water natural frequency ( f , ), to form a normalized velocity (;*/f* ), also called &&true'' reduced velocity in recent studies, we "nd an excellent collapse of data for a set of response amplitude plots, over a wide range of mass ratios m*. Such a collapse of response plots cannot be predicted a priori, and appears to be the "rst time such a collapse of data sets has been made in free vibration. The response branches match very well the Williamson}Roshko (Williamson & Roshko 1988) map of vortex wake patterns from forced vibration studies. Visualization of the modes indicates that the initial branch is associated with the 2S mode of vortex formation, while the Lower branch corresponds with the 2P mode. Simultaneous measurements of lift and drag have been made with the displacement, and show a large ampli"cation of maximum, mean and #uctuating forces on the body, which is not unexpected. It is possible to simply estimate the lift force and phase using the displacement amplitude and frequency. This approach is reasonable only for very low m*.

579Two fundamental characteristics of the low-Reynolds-number cylinder wake, which have involved considerable debate, are first the existence of discontinuities in the Strouhal-Reynolds number relationship, and secondly the phenomenon of oblique vortex shedding. The present paper shows that both of these characteristics of the wake are directly related to each other, and that both are influenced by the boundary conditions at the ends of the cylinder, even for spans of hundreds of diameters in length. It is found that a Strouhal discontinuity exists, which is not due to any of the previously proposed mechanisms, but instead is caused by a transition from one oblique shedding mode to another oblique mode. This transition is explained by a change from one mode where the central flow over the span matches the end boundary conditions to one where the central flow is unable to match the end conditions. In the latter case, quasi-periodic spectra of the velocity fluctuations appear; these are due to the presence of span wise cells of different frequency. During periods when vortices in neighbouring cells move out of phase with each other, 'vortex dislocations' are observed, and are associated with rather complex vortex linking between the cells. However, by manipulating the end boundary conditions, parallel shedding can be induced, which then results in a completely continuous Strouhal curve. It is also universal in the sense that the oblique-shedding Strouhal data (S 11 ) can be collapsed onto the parallel-shedding Strouhal curve (S 0 ) by the transformation, S 0 = S 11 jcos (), where () is the angle of oblique shedding. Close agreement between measurements in two distinctly different facilities confirms the continuous and universal nature of this Strouhal curve. It is believed that the case of parallel shedding represents truly two-dimensional shedding, and a comparison of Strouhal frequency data is made with several two-dimensional numerical simulations, yielding a large disparity which is not clearly understood. The oblique and parallel modes of vortex shedding are both intrinsic to the flow over a cylinder, and are simply solutions to different problems, because the boundary conditions are different in each case.

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 ↔ 2P vortex wake modes, and a corresponding switch in vortex shedding timing. Across the second mode transition, there is a jump in 'total phase', φ 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: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.

Although there are a great many papers dedicated to the problem of a cylinder vibrating transverse to a fluid flow ($Y$-motion), there are almost no papers on the more practical case of vortex-induced vibration in two degrees of freedom ($X,Y$ motion) where the mass and natural frequencies are precisely the same in both $X$- and $Y$-directions. We have designed the present pendulum apparatus to achieve both of these criteria. Even down to the low mass ratios, where $m^*\,{=}\,6$, it is remarkable that the freedom to oscillate in-line with the flow affects the transverse vibration surprisingly little. The same response branches, peak amplitudes, and vortex wake modes are found for both $Y$-only and $X,Y$ motion. There is, however, a dramatic change in the fluid–structure interactions when mass ratios are reduced below $m^*\,{=}\,6$. A new amplitude response branch with significant streamwise motion appears, in what we call the ‘super-upper’ branch, yielding massive amplitudes of 3 diameters peak-to-peak ($A^*_Y \,{\sim}\, 1.5$). We discover a corresponding periodic vortex wake mode, comprising a triplet of vortices being formed in each half-cycle, in what we define as a ‘2T’ mode. We qualitatively interpret the principal vortex dynamics and vortex forces which yield a positive rate of energy transfer ($\dot{e}_V$) causing the body vibration, using the following simple equation: \[\dot{e}_V = 2 {\Gamma}^* U^*_V \skew3\dot{Y}\] where $\Gamma^*$ is vortex strength, $U^*_V$ is the speed downstream of the dominant near-wake vorticity, and $\skew3\dot{Y}$ is the transverse velocity of the body. This simple approach suggests that the massive amplitude of vibration for the 2T mode is principally attributed to the energy transfer from the ‘third’ vortex of each triplet, which is not present in the lower-amplitude 2P mode. We also find two low-speed streamwise vibration modes, which is not unexpected, since they correspond to the first and second excitation modes of vibration for flexible cantilevers. By considering equations of motion for the two degrees of freedom, we find a critical mass, $m^*_{\hbox{\scriptsize\it crit}} \,{=}\, 0.52$, similar to recent $Y$-only studies, below which the large-amplitude vibrations persist to infinite flow velocity. We show that the critical mass $m^*_{\hbox{\scriptsize\it crit}}$ is the same for the $X$- and $Y$-directions, which ensures that the shapes of $X, Y$ trajectories can retain their form as the velocity becomes large. The extensive studies of vortex-induced vibration for $Y$-only body motions, built up over the last 35 years, remain of strong relevance to the case of two degrees of freedom, for $m^* \,{>}\, 6$. It is only for ‘small’ mass ratios, $m^* \,{<}\, 6$, that one observes a rather dramatic departure from previous results, which would suggest a possible modification to offshore design codes.

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