The influence of taper ratio on vortex-induced vibration of tapered cylinders in the crossflow direction
Introduction
Vortex-induced vibration (VIV) of a circular cylinder with a constant diameter placed normal to the flow has been studied extensively for a flexibly-mounted rigid cylinder (Bearman, 1984, Sarpkaya, 2004, Vandiver, 2012, Williamson and Govardhan, 2004). Here, VIV of a tapered cylinder – a cylinder in which the cross-sectional area varies continuously along its length – is considered.
The wake of a tapered cylinder placed in uniform flow has been studied widely for a fixed cylinder. Gaster, 1969, Gaster, 1971 conducted the first series of experiments to study the wake of a fixed tapered cylinder placed in uniform flow. His work was continued by Papangelou (1992), Piccirillo and Van Atta (1993), and later Hsiao and Chiang (1998). Valles et al. (2002) used computer simulations and reproduced the laminar vortex shedding behind a linearly tapered cylinder observed experimentally by Piccirillo and Van Atta (1993). Several studies have used direct numerical simulations to understand the vortex shedding behind tapered cylinders at low Reynolds numbers (Narasimhamurthy et al., 2009, Parnaudeau et al., 2007, Provansal and Monkewitz, 2006) and some have used stereo-PIV (Visscher et al., 2011). Vortex dislocations, cellular shedding and oblique shedding are the major phenomena discussed in the studies of a fixed tapered cylinder placed in flow.
The dynamics of an oscillating tapered cylinder, however, have received less attention. Techet et al. (1998) conducted a series of tests on a tapered cylinder with a taper ratio of 40:1 where the taper ratio was defined as the ratio of the cylinder length to the difference between the maximum and minimum diameters of the cylinder. The cylinder was forced to oscillate in the transverse direction and towed in a towing tank. They observed both 2S (two single vortices per cycle of oscillation) and 2P (two pairs of vortices per cycle of oscillation) vortex shedding in the wake of the cylinder as well as a new “hybrid” shedding, which consisted of 2S shedding in the range of the cylinder with larger diameter, and 2P shedding in the range of the cylinder with smaller diameter. Hover et al. (1998) continued this work by conducting a series of experiments on a flexibly-mounted tapered cylinder with the same taper ratio, and confirmed the validity of the forced VIV tests to predict the amplitude of oscillations and the phase between the lift and the displacement. They also found that the lock-in range in the tapered cylinder started at a slightly lower value for the nominal reduced velocity (reduced velocity defined based on the natural frequency of the structure).
Balasubramanian et al. (2001) studied VIV of a pivoted tapered circular cylinder placed either in a uniform or sheared flow, and noted the differences of the resulting lock-in range with the case of a pivoted uniform cylinder. Their experimental investigation revealed the importance of the relative orientation of the sheared flow gradient with respect to the orientation of the tapered cylinder. They also related the different amplitudes of vibrations to the mass distribution of the tapered cylinder along its length. Since the tapered cylinder in their study was hinged with unequal mass distribution along the length, the results were found to be very sensitive to the relative direction of the cylinder with respect to the incoming flow.
Zeinoddini et al. (2013) conducted VIV tests on tapered cylinders with two different taper ratios. A comprehensive study of the influence of the taper ratio was not possible in their tests, since the cylinders with two taper ratios had two different mass ratios. They observed a wider lock-in range for the tapered cylinders, compared with the uniform cylinders. They also tested the tapered cylinders with a pure inline motion and did not observe any significant difference in the amplitude of oscillations compared with the case of a uniform cylinder.
In the present paper, a series of tests are discussed on four tapered cylinders, placed in flow and free to oscillate in the crossflow direction (i.e., in the transverse direction). Three of the cylinders had linear taper, in which the diameter of the cylinders varied linearly from one end to the other end, with three different taper ratios, and the fourth cylinder had a nonlinear taper. A series of tests on a uniform cylinder were conducted as well, to have a basis for comparison. To be able to draw conclusions on the influence of the taper ratio, the mass ratio and the structural damping were kept constant in all these cases. Tests were conducted with and without end plates and with the tapered cylinder in two orientations (smaller diameter in the top or in the bottom) to study the influence of the boundary conditions on the resulting response of the cylinder.
Section snippets
Experimental set-up
Four tapered cylinders were considered in the experiments. Three of them had a linear taper with taper ratios of 29:1 (cylinder L1), 17:1 (cylinder L2) and 10:1 (cylinder L3), where the taper ratio is defined as τ=(L/Dmax–Dmin), in which L is the cylinder length, and Dmax and Dmin are the maximum and minimum diameters, respectively. The fourth cylinder had a nonlinear taper (Cylinder N), in which the cylinder’s diameter, D, varied along its length from Dmin to Dmax as z(D)=0.003D2–0.0196D
Amplitude and frequency of oscillations
The first series of tests were conducted using all three cylinders of Fig. 2 with a linear taper (cylinders L1, L2 and L3). Each cylinder was placed vertically in the test-section in a way that its smaller diameter was placed in the top, close to the water surface. A circular end-plate with a radius of 5.5 cm and a thickness of 2 mm was attached to the lower end of the cylinder, as suggested by Morse et al. (2008). The edges of the end-plate were streamlined to avoid flow separation. The mass
A nonlinearly tapered cylinder
Besides the linearly tapered cylinders discussed in the previous section, a nonlinearly tapered cylinder, in which the cylinder׳s diameter varied along its length from Dmin to Dmax as z(D)=0.003D2–0.0196D+0.201, was used to study the influence of the nonlinear taper on the resulting crossflow VIV. The nonlinear cylinder was designed such that the average value of its maximum and minimum diameters, (Dmax+Dmin)/2, was equal to the average diameter of the linearly tapered cylinders discussed in
Inverted configurations
In the tests discussed in the previous sections, the tapered cylinder was placed in the test section of the water tunnel such that its upper end had a smaller diameter and its lower end, a larger diameter. The tests were repeated with inverted configurations (the larger diameter in the top) in order to study if the boundary conditions were affecting the observed vortex shedding and the corresponding oscillations. Fig. 10 shows the lock-in ranges for the tapered cylinders in their inverted
The influence of the lower end
In order to study the end effects, the response of the system was studied for cylinders with a free lower end (i.e., no end-plate was attached to the cylinder). Fig. 13 shows the lock-in range for a uniform cylinder with and without the end-plate for a mass ratio of m⁎=7.5. A dummy mass was added to the moving part of the cylinder with no end plate, so that the mass ratio of the two cases remained at a constant value of m⁎=7.5. As shown in Fig. 13, adding the end-plate resulted in two distinct
Conclusions
Vortex-induced vibration of tapered cylinders free to oscillate in the crossflow direction was studied experimentally. The previous studies on tapered cylinders were focused on large taper ratios (cases closer to a uniform cylinder). By considering smaller taper ratios (cases with larger deviations from a uniform cylinder) in the present work, the influence of the taper ratio was more noticeable in the results. Three linearly and one nonlinearly tapered cylinders were considered. For linearly
Acknowledgements
The financial support provided by the Armstrong Fund for Science is greatly appreciated.
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