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Mass ratio effect on vortex induced vibration of a flexibly mounted circular cylinder, an experimental study
International Journal of Marine Energy 16 (2016) 1–11
Contents lists available at ScienceDirect
International Journal of Marine Energy journal homepage: www.elsevier.com/locate/ijome
Mass ratio effect on vortex induced vibration of a flexibly mounted circular cylinder, an experimental study Alireza Modir ⇑, Mohsen Kahrom, Anoshirvan Farshidianfar Department of Mechanical Engineering, Ferdowsi University of Mashhad, Iran
a r t i c l e
i n f o
Article history: Received 2 December 2015 Revised 12 April 2016 Accepted 3 May 2016 Available online 6 May 2016 Keywords: Mass ratio Vortex induced vibration Circular cylinder Towing-tank
a b s t r a c t The effect of mass ratio (m⁄ = the mass of oscillating body/the mass of displaced fluid) on vortex induced vibration of an elastically mounted rigid circular cylinder over a wide range of Reynolds numbers (1.7 104 < Re < 7 104) in a high damping system is studied in this paper. The cylinder is limited to a transverse oscillation and is carried inside a 14 m long water channel for constant velocities. The amplitude of response depends on various parameters such as mass ratio, damping ratio, natural frequency of system and Reynolds number. Here we considered three different circular cylinders with the same diameter and length, but distinct masses (m⁄ = 1.6, 2.3 and 3.4). The experiments carried out in a towing-tank water channel and the following results achieved: the peak amplitude of oscillation principally depends on the mass ratio and it increases with decrease of m⁄. For systems with constant mass-damping parameters, by reducing the mass ratio, the maximum amplitude of oscillation remains constant, while the range of synchronization increases considerably. Higher Reynolds numbers in our experiments led to reach the same maximum amplitude of oscillation as some of the previous studies (A⁄ = 1), even with higher mass-damping parameter (m⁄f) in our system. The high damping in our experiments, resulted in disappearance of the lower branch in the amplitude response graphs. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Vortex-induced vibration of the cylinder has been studied extensively during the past fifty years. The study on response of an elastically mounted circular cylinder free to vibrate transverse to the flow has been started since the early work of Feng [1]. The range of problems caused by VIV has led to a large number of numerical and experimental studies on this subject, including various comprehensive review articles [2–5]. VIV was known as a destructive phenomenon and many researchers tried to reduce and diminish this harmful energy that causes failure in engineering structures. Recently, Bernitsas and his group at the University of Michigan Marine Renewable Energy Lab came up with the idea of harnessing the water current energy by utilizing vortex induced vibration [6–8]. They invented a device and called that VIVACE (Vortex Induced Vibration for Aquatic Clean Energy). This device generates clean hydrokinetic energy from slow-moving water currents by using the oscillation of the cylinder under VIV. Raghavan and Bernitsas [8], studied the Reynolds number effect on vortex induced vibration of a rigid mounted circular cylinder for Reynolds regimes right before transition from laminar to turbulent flow. They found that VIV is significantly different between the TrSL2 and TrSL3 flow regimes. ⇑ Corresponding author. E-mail address: [email protected] (A. Modir). http://dx.doi.org/10.1016/j.ijome.2016.05.001 2214-1669/Ó 2016 Elsevier Ltd. All rights reserved.
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As reported in [9], two types of response are possible for a free vibration VIV system. The first one is a small-amplitude and two branch response which corresponds to the systems with high mass-damping parameters (m⁄f). Most of these systems used air as their working fluid [10]. These branches were named as the ‘‘initial” and the ‘‘lower” branches. The second form of response is a large-amplitude, three-branch response that was believed to result from low mass-damping. Most of experiments with three branch response, used water as their working fluid [10]. The three branches were called the ‘‘initial,” ‘‘upper,” and ‘‘lower” branches. Govardhan and Williamson [11] suggested the condition A/D = 0.6, the limit to divide the regimes of three-branch and two-branch response. In Fig. 1, Feng’s minimum damping case and Khalak and Williamson’s case are compared [5]. The maximum amplitude of oscillation that can be achieved by a particular system is a subject that attracts investigators for studying different aspects of VIV. Griffin [12] for the first time collected the other investigators data and showed the relationship between maximum amplitude of oscillation for a system and a mass-damping parameter called reduced damping (SG = 2p3St2(m⁄f)). This graph is called Griffin plot and can be seen in Fig. 2. He observed that as the reduced damping increases, the related maximum amplitude of oscillation decreases, and for high mass ratio or high damping the oscillation stops. Govardhan and Williamson [13] studied the effect of Reynolds number as another effective parameter on amplitude of oscillation and changed the Griffin plot based on Reynolds number. They could find a better coverage for different experiments. Raghavan and Bernitsas [8] showed that this plot could be valid just for experiments done in the TrSL2 flow regime and their data, which was accomplished for Reynolds numbers in TrSL3 and TrBL0 regimes, was two times higher than the values obtained by Govardhan and Williamson [11]. Khalak and Williamson [9] have shown that the response of the cylinder strongly depends on the mass-damping parameter (m⁄f). They observed that for a constant value of m⁄f, by reducing mass ratio the maximum amplitude of oscillation remains almost constant, but the synchronization range increases. Govardhan and Williamson [13] studied the critical mass of a vertical cylinder in VIV. They showed that for m⁄ < 0.54, the body will oscillate over an ultimate regime of normalized velocities (U⁄) and named that, the critical mass ratio. Mass ratio plays an important role for practical applications. The goal of our research is to study the significant effect of mass ratio on vortex induced vibration of a horizontally mounted circular cylinder at high Reynolds numbers and high values of damping. 2. Physical mechanism Vortex Induced occurs when a current flows over a bluff body. For real viscous fluid flows, there will be a remarkable boundary layer on the surface of the body. For Reynolds number greater than 40, separation of the boundary layer will occur [8]. This separated layer will cause fluid rotation and this rotation causes formation of vortices, which are then shed from the back of the body and travel down the wake. During this process, vortices form and shed on the downstream side of bluff bodies in a current and induce periodic forces on the body. Fig. 3 shows this process on a flexibly mounted circular cylinder, which cause the cylinder to vibrate transverse to the flow direction. The equation of motion that normally used by many investigators to represent VIV of a circular cylinder oscillating in transverse Y direction (normal to the flow) is as below [11].
€ þ cy_ þ ky ¼ F my
ð1Þ
where m = mass of oscillating body, c = structural damping, k = spring stiffness and F = fluid force in the transverse direction. During the synchronization, a suitable approximation for force and cylinder oscillation can be given as follows:
FðtÞ ¼ F 0 sinðxt þ /Þ
Fig. 1. Amplitude response plot versus reduced velocity for two of the previous studies [5].
ð2Þ
A. Modir et al. / International Journal of Marine Energy 16 (2016) 1–11
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Fig. 2. Griffin plot which shows the maximum amplitude of oscillation versus mass-damping parameter [5].
yðtÞ ¼ y0 sinðxtÞ
ð3Þ
where x = 2pf and f = oscillation frequency. By considering that our study is basically experimental, the numerical calculations would not be considered in detail. 3. Dimensionless parameters 3.1. Reynolds number The first important parameter, and perhaps the most important parameter for studying VIV is Reynolds number (Re), which is defined by Eq. (4) (U is the free stream velocity, D is the cylinder diameter, and m is the kinematic viscosity of the fluid). Zdravkovich [14] classified the flow over a cylinder into 15 flow regimes. The range of 1 104 –2 104 < Re < 1 105–2 105, is named TrSL3 and our experiments also lie in this range. Raghavan and Bernitsas [8] compared the flow properties of TrSL2 and TrSL3 and stated that VIV behavior is significantly different in these two regimes.
Re ¼
UD
ð4Þ
m
3.2. Strouhal number The nondimensional frequency at which the vortices are shed into the wake is known as the Strouhal number (St) and is defined by Eq. (5) (fvs is the vortex shedding frequency for a stationary cylinder).
Fig. 3. Vortex induced vibration phenomenon.
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f D St ¼ v s U
ð5Þ
The relation between Reynolds number and Strouhal number for circular cylinders is shown in Fig. 4. In a wide range of Reynolds numbers, the Strouhal number varies very little, and can be considered constant. For the entire range of about 103 < Re < 105, the Strouhal number is nearly 0.2 for smooth surfaces [15], which corresponds very well to the range of Reynolds number in our experiments. Therefore, St will be taken as a constant value in any calculation. So the vortex shedding frequency (fvs) is then calculated from Eq. (6).
f vs ¼
StU D
ð6Þ
3.3. Reduced velocity (U⁄) The non-dimensional parameter used in many papers for indicating vibration amplitude, is the reduced velocity U⁄, given by Eq. (7) (fn is the natural frequency of the system in water).
U ¼
U f nD
ð7Þ
3.4. Amplitude ratio (A⁄) Normalized oscillation amplitude used in many papers, is given as A⁄ = amplitude/diameter. 4. Important parameters for the analysis of VIV 4.1. Damping ratio (f) Damping ratio is a measure for the ability of the structure to dissipate energy in a cycle of vibration. The damping of the system includes structural (frictional) damping and fluid damping. Damping ratio can be calculated by performing a free decay test, by giving an initial displacement to the cylinder and measuring the two successive peaks (yn and yn+1) of the cylinder in the water. Hence, the damping ratio can be calculated from Eq. (5).
f¼
1 y ln n 2p ynþ1
ð8Þ
4.2. Natural frequency in water Natural frequency in water is defined as bellow
1 fn ¼ 2p
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k mosc þ ma
ð9Þ
The added mass is ma = CAmd, where md = qwpD2L/4 and CA is the ideal added mass coefficient that is equal 1 for a circular cylinder [11].
Fig. 4. St–Re relationship for circular cylinders [15].
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4.3. Mass ratio (m⁄) The mass ratio is defined as the total mass of all oscillating parts plus 1/3 of the spring mass divided by the displaced fluid mass (md) [8].
m ¼
mosc mosc ¼ md qw p4 D2 L
ð10Þ
5. Lock-in Lock-in is like linear resonance in which the vibration amplitude increases as the vortex shedding frequency (fs) gets close to the natural frequency of the system (fn). The phenomenon of lock-in traditionally means that the frequency ratio (f⁄ = fosc/ fn,water) remains close to unity [10]. Accordingly, lock-in will occur when the reduced velocity (U⁄) becomes approximately equal to the inverse of the Strouhal number (1/St) which is equal to 5. In the definition given by Sarpkaya [16], during lock-in, fluid force frequency should be equal to oscillation frequency. In systems with low mass ratio, the cylinder oscillates at a frequency higher than the natural frequency of the system. The deviation of f⁄ from unity in systems under VIV is shown in [3] and [17,18]. Lock-in is a highly non-linear phenomenon, and can occur over a wide range of shedding frequencies. Lock-in also has been described as a self-limiting and self-governing phenomenon. The cylinder vibrations affect the vortex shedding procedure, and vice versa. It is known as self-limiting, due to the fact that as the cylinder displacement rises, the vortex shedding is getting weak, and accordingly decreases more motion [7]. Synchronization is defined as the matching of the vortex shedding frequency (fvs) with oscillation frequency (fosc) [18]. In many papers ‘‘Lock-in” and ‘‘Synchronization” are used interchangeably. It worth to mention that ‘‘Lock-in” and ‘‘Synchronization” have different meanings. ‘‘Lock-in” occurs when the vortex shedding frequency breaks from the Strouhal relationship and fits the natural frequency of the system. On the other side, ‘‘Synchronization” happens when vortex shedding frequency breaks from Strouhal relationship but matches a frequency that is not essentially the natural frequency of the system. Consequently, lock-in is a special case of synchronization [19]. 6. Experimental set-up Different aspects of VIV including the range of synchronization, branches of response within the synchronization, effect of mass ratio on maximum amplitude of oscillation and frequency response, are studied in this paper. Experiments are carried out in a towing-tank water channel which has 14 m length, with a cross section of 50 cm width and 60 cm depth. The purpose of the towing-tank water channel is to facilitate the study of the flow over a flexibly mounted circular cylinder in a wide range of velocities. The experimental set-up is consist of a horizontal cylinder suspended from a carriage, which rides inside the calm water channel. Three cylinders with a constant geometry but different masses are used here. All cylinders have a diameter of 6 cm and length of 38 cm with smooth surfaces. Cylinders are made from stainless steel, compressed plastic and iron, with mass ratios of 1.64, 2.26 and 3.39 respectively (Fig. 5). The weight of the overall oscillating mechanism that holds the cylinder and oscillates with it, is 1.21 kg. Aspect ratio (Length/Diameter) of all cylinders is 6.3, and the blockage ratio at every section of the channel is 11.8%. The main properties of the three different cases used in this paper are listed in Table 1. The cylinders are limited to oscillate transversely to the carriage movement by using 2 cylindrical bearings, as showing in Fig. 6. In performing the experiments, the stiffness of the springs are kept constant at 570 N/m, and the relative flow speed (the carriage speed) varied from 0.38 m/s to 1.5 m/s, yielding Reynolds number from 17,000 to 70,000. By employing an electronic inverter connected to a three-phase electromotor, the carriage speed can be changed in a broad range of velocities. The current effect is created by towing the cylinder at a constant speed inside the calm water. The relative flow passing the cylinder is equivalent with the condition that the cylinder is fixed and the water flows over it. Since tests are done in a calm water channel with sufficient time between runs, the free stream turbulence was reduced to negligible levels. The displacement of the cylinder is measured by using an advanced digital camera, recording 30 frames per second, and a ruler which allows recording the position of the cylinder during tests. 7. Results and discussion By performing the free-decay test in water, the damping ratio for three different systems obtained. Fig. 7 shows a sample of damping calculation test for an initial displacement of 40 mm. By considering that there is a metal–metal contact between bearings and the side struts, the damping associated with the bearing system is moderately high which result in high values of mass-damping parameter in the three cases reported here. Due to the facilities employed in our experiments, a wide range of reduced velocities obtained (3 < U⁄ < 11). This range is large enough to identify the lock-in region, where the higher amplitudes are observed. At lower U⁄ the response is clearly quasiperiodic, while for the slightly higher U⁄, the response becomes suddenly more periodic. Fig. 8 illustrates the time trace of a cylinder with m⁄ = 2.26 and U⁄ = 6.5 for the first four seconds.
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Fig. 5. The three different cylinders used in the present study.
Table 1 VIV model particulars. Case
mcylinder
mosc
md
(m⁄)
(f)
K (N/m)
fn
1 2 3
0.540 1.185 2.400
1.750 2.395 3.610
1.072 1.072 1.072
1.64 2.26 3.39
0.061 0.053 0.059
570 570 570
2.26 2.03 1.75
In our VIV experiments performed for three different mass ratios, maximum amplitude ratio about 1 was measured for Reynolds numbers in the range of 1.7 104 < Re < 7 104. Figs. 9 and 10 display the amplitude ratio (A/D) versus reduced velocity (U⁄) and Reynolds number (Re), for different mass ratios. Initial, Upper and lower branches are shown in Fig. 9. The amplitude of oscillation shows a decreasing trend with increase in mass ratio of the system. Based on the definition expressed in [11], it can be concluded that the upper branch disappeared for the case of m⁄ = 3.39 (A/D < 0.6). It clearly can be noticed from Fig. 10 that the amplitude of oscillation and the range of synchronization in the upper branch increase with rise in Reynolds number. As it can be observed from this figure, the amplitude of oscillation versus Reynolds number fit in one curve in the initial branch for different cylinders. It is clearly shown in Fig. 10 that by decreasing m⁄, the range of synchronization expands and shifts to the higher Reynolds numbers which shows the dependence of VIV on mass ratio. Based on the definition of Govardhan and Williamson [11], A/D = 6 is the demarcation for the three branch
Fig. 6. Main components of the VIV test model, (1) cylinder, (2) side strut, (3) linear bearings, (4) springs, (5) wheels.
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Fig. 7. A sample of damping ratio calculation, derived from the free-vibration decay test in water.
Fig. 8. A sample time-trace plot for the first four seconds of the test.
response, it was expected to observe these branches in our graphs. It can be seen by rising the reduced velocity, the amplitude ratio jumps from initial branch to the upper branch and it suddenly drops to desynchronization without experiencing the lower branch which is not a common VIV character. Raghavan and Bernitsas [8] also observed that for their high massdamping case, the lower branch disappeared. They witnessed a drop from synchronization to desynchronization without any hump in their amplitude response. It is essential to understand how the maximum amplitude of oscillation (A⁄max) changes as a function of m⁄ and f. Williamson and Govardhan [5], mentioned that the product of ‘‘m⁄f”, is the parameter which controls the maximum amplitude of oscillation. The increase in the damping can be counteracted by decreasing the mass ratio [9]. In Fig. 11 the maximum amplitude of oscillations for the three cylinders are plotted as a function of the Skop-Griffin Parameter (SG). Our results approximately fit to the classical Griffin plot, which shows the maximum amplitude of oscillation as a function of the mass-damping parameter.
7.1. Comparison In this part of paper we compare our results with three of the best-known studies on VIV of a flexibly mounted circular cylinder. In Fig. 12, the experiments done in [1,17,20], and our results are plotted in the same graph. The characteristics of these four experiments, including Reynolds number, mass ratio, mass-damping parameter ((m⁄ + CA)f) and maximum ampli-
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Fig. 9. Amplitude ratio versus reduced velocity for different mass ratios.
Fig. 10. Amplitude ratio versus Reynolds number for different mass ratios.
Fig. 11. Result of present study in the typical linear-log form of Griffin plot.
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Fig. 12. The comparison of the experiments done by Feng [1], Khalak and Williamson [17], Govardhan and Williamson [20] and the present study.
Table 2 The experimental characteristic of three well-known investigators and the present experiment. Investigator Feng [1] Khalak et al. [17] Govardhan et al. [20] Present study
Reynolds 3
4
6.0 10 –2.3 10 5.0 103–1.6 104 2.9 103–1.9 104 1.2 104–6.8 104
m⁄
(m⁄ + CA)f
A/D
250 2.4 1.19 1.64
0.36 0.013 0.011 0.16
0.55 0.94 1.0 1.0
Fig. 13. Nondimentional frequency of response, f⁄, versus nondimentional velocity, U⁄.
tude of oscillation are listed in Table 2. As stated above, added mass coefficient (CA) is assumed to have a value of unity for a circular cylinder [13]. In Fig. 12, we may compare our amplitude response plot for m⁄ = 1.64 and f = 0.061 with the response measured by Feng [1], in whose experiments m⁄ = 248 and f = 0.0014. Since the present results have lower mass-damping than those of Feng, (less than half of his value), we find much greater amplitudes, with nearly twice the maximum amplitude of what Feng could achieve. As one can observe in Fig. 12, the range of synchronization and the maximum amplitude of oscillation in our experiments are very close to the one measured by Khalak and Williamson [17], while the mass-damping parameter in our experiments is 12 times higher. One of the reasons for accomplishing the same maximum amplitude in these two experiments can be explained due to the higher Reynolds numbers and consequently higher lift coefficient [8] in our case.
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Fig. 14. Oscillation frequency of cylinders versus velocity.
By comparing the results of [17] with [20], it can be concluded that by keeping constant the mass ratio parameter ((m⁄ + CA)f) for almost the same Reynolds number, maximum amplitude of oscillation stays constant but the synchronization range increases by decreasing the mass ratio of the system. This explanation can be an evidence of disappearance of lower branch in our experiment. 7.2. Frequency response The frequency responses of the three cylinders are shown in Fig. 13. In this figure, the ratio of the oscillation frequency to the natural frequency in water (f⁄ = fosc/fn,water) is plotted versus reduced velocity. The Strouhal frequency (inclined line) is also shown for comparison. In the initial branch of oscillation, cylinders oscillate at lower frequency than the natural frequency of the system and by increasing the reduced velocity throughout the upper branch, the nondimentional frequency deviates slowly from unity, reaching its final value near 1.3 for U⁄ = 11. This does not correspond to either Strouhal frequency of a stationary cylinder (f⁄ = 0.2U⁄) or the natural frequency of the system (f⁄ = 1), but lies between them. At reduced velocities close to 7, the cylinders oscillate at the natural frequency of the system which is the definition of lock-in. By referring to Fig. 9, it can be observed that cylinders oscillate at the maximum amplitude of oscillation in this value of reduced velocity. The frequency of oscillation versus current velocity (U) for all cylinders are shown in Fig. 14. It is interesting to mention that regardless the mass ratio of the cylinders, all of them oscillate at the same frequency for a specific fluid velocity and the frequency response curves collapse into one curve in the synchronization regime. By increasing fluid velocity, it can be observed that the oscillation frequency follows an increasing trend for all cylinders. 8. Conclusions The effect of mass ratio on vortex-induced vibration of an elastically mounted rigid circular cylinder was tested in a towing-tank water channel for three different cylinders with the same geometry but different masses. The mass ratio of the cylinders were m⁄ = 1.64, 2.26 and 3.39, while the damping ratio and spring stiffness were kept constant. The experiments were done in a wide range of Reynolds numbers (1.7 104 < Re < 7 104) in a towing-tank water channel. All our results for high damping systems show strong depends of VIV on mass ratio. The maximum amplitude of oscillation obtained during the experiments was considerably close to the results obtained in [17], while our mass-damping parameter was noticeably higher. Different amplitude response branches were clearly shown in different plots and the following conclusions can be made: (1) The maximum amplitude of oscillation and the range of synchronization strongly depend on mass ratio. (2) As m⁄ decreases, the maximum amplitude of oscillation increases and the range of synchronization expands and shifts to higher Reynolds numbers. (3) The range of synchronization in upper branch increases with rise in Reynolds number. (4) The upper branch disappeared (A⁄ < 0.6) for the highest mass ratio case (m⁄ = 3.39). (5) In our experiments with high damping parameter, the amplitude of oscillation in upper branch fell sharply to the desynchronization without experiencing a detectable lower branch.
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(6) The nondimentional frequency stays close to unity in upper branch and deviates slowly from 1 with increasing the reduced velocity. In lower branches, the nondimensional frequency of oscillation was greater than unity, but lower than the Strouhal frequency. (7) All three cylinders oscillated at the same frequency for different fluid velocities. (8) By comparing our results with [17,20], it can be concluded that maximum amplitude of oscillation principally dependents on the mass-damping parameter (m⁄f), whereas the regime of synchronization (measured by the range of U⁄), for a given value of m⁄f, is dependent primarily on the mass ratio, only.
References [1] C.C. Feng, The Measurement of Vortex-Induced Effects in Flow Past a Stationary and Oscillating Circular Cylinder and d-Section Cylinders (Master’s thesis), University of British Columbia, Vancouver, Canada, 1968. [2] T. Sarpkaya, Vortex-induced oscillations, J. Appl. Mech. 46 (1979) 241–258. [3] P.W. Bearman, Vortex shedding from oscillating bluff bodies, Annu. Rev. Fluid Mech. 16 (1984) 195–222. [4] C.H.K. Williamson, Vortex dynamics in the cylinder wake, Annu. Rev. Fluid Mech. 28 (1996) 477–539. [5] C.H.K. Williamson, R. Govardhan, Vortex-induced vibrations, Annu. Rev. Fluid Mech. 36 (2004) 413–455. [6] M.M. Bernitsas, K. Raghavan, Y. Ben-Simon, E.M. Garcia, VIVACE (Vortex Induced Vibration Aquatic Clean Energy): a new concept in generation of clean and renewable energy from fluid flow, J. Offshore Mech. Arct. Eng. ASME Trans. 130 (4) (2008) 041101–041115. [7] M.M. Bernitsas, Y. Ben-Simon, K. Raghavan, E.M.H. Garcia, The VIVACE converter: model tests at high damping and Reynolds Number around 105, J. Offshore Mech. Arct. Eng. Trans. ASME 131 (2009) 1. [8] K. Raghavan, M.M. Bernitsas, Experimental investigation of Reynolds number effect on vortex induced vibration of rigid cylinder on elastic supports, Ocean Eng. 38 (2011) 719–731. [9] A. Khalak, C.H.K. Williamson, Motions forces and mode transitions in vortex induced vibrations at low mass-damping, J. Fluids Struct. 13 (1999) 813– 851. [10] J.T. Klamo, Effects of Damping and Reynolds Number on Vortex-Induced Vibrations (Ph.D. thesis), California institute of Technology, 2007. [11] R. Govardhan, C.H.K. Williamson, Defining the ‘modified Griffin plot’ in vortex-induced vibration: revealing the effect of Reynolds number using controlled damping, J. Fluid Mech. 561 (2006) 147–180. [12] O.M. Griffin, R.A. Skop, S.E. Ramberg, The resonant, vortex-excited vibrations of structures and cable systems, in: Annual Offshore Technology Conference, May, Houston, TX, 1975. [13] R. Govardhan, C.H.K. Williamson, Critical mass in vortex-induced vibration of a cylinder, Eur. J. Mech. Fluids 23 (2004) 17–27. [14] M.M. Zdravkovich, Flow Around Circular Cylinders Volume 1: Fundamen-tals, Oxford Science Publications, 1997. [15] R.D. Blevins, Flow-Induced Vibration, second ed., Van Nostrand Reinhold, New York, 1990. [16] T. Sarpkaya, Fluid forces on oscillating cylinders, J. Waterw. Port Coastal Ocean Div. 104 (1978) 275–290. [17] A. Khalak, C.H.K. Williamson, Fluid forces and dynamics of a hydroelastic structure with very low mass and damping, J. Fluids Struct. 11 (1997) 973– 982. [18] A. Khalak, C.H.K. Williamson, Investigation of relative effects of mass and damping in vortex-induced vibration of a circular cylinder, J. Wind Eng. Ind. Aerodyn. 69 (1997) 341–350. [19] J.T. Klamo, A. Leonard, A. Roshko, The effects of damping on the amplitude and frequency response of a freely vibrating cylinder in cross-flow, J. Fluids Struct. 22 (2006) 845–856. [20] R. Govardhan, C.H.K. Williamson, Modes of vortex formation and frequency response of a freely vibrating cylinder, J. Fluid Mech. 420 (2000) 81–930.
Contents lists available at ScienceDirect
International Journal of Marine Energy journal homepage: www.elsevier.com/locate/ijome
Mass ratio effect on vortex induced vibration of a flexibly mounted circular cylinder, an experimental study Alireza Modir ⇑, Mohsen Kahrom, Anoshirvan Farshidianfar Department of Mechanical Engineering, Ferdowsi University of Mashhad, Iran
a r t i c l e
i n f o
Article history: Received 2 December 2015 Revised 12 April 2016 Accepted 3 May 2016 Available online 6 May 2016 Keywords: Mass ratio Vortex induced vibration Circular cylinder Towing-tank
a b s t r a c t The effect of mass ratio (m⁄ = the mass of oscillating body/the mass of displaced fluid) on vortex induced vibration of an elastically mounted rigid circular cylinder over a wide range of Reynolds numbers (1.7 104 < Re < 7 104) in a high damping system is studied in this paper. The cylinder is limited to a transverse oscillation and is carried inside a 14 m long water channel for constant velocities. The amplitude of response depends on various parameters such as mass ratio, damping ratio, natural frequency of system and Reynolds number. Here we considered three different circular cylinders with the same diameter and length, but distinct masses (m⁄ = 1.6, 2.3 and 3.4). The experiments carried out in a towing-tank water channel and the following results achieved: the peak amplitude of oscillation principally depends on the mass ratio and it increases with decrease of m⁄. For systems with constant mass-damping parameters, by reducing the mass ratio, the maximum amplitude of oscillation remains constant, while the range of synchronization increases considerably. Higher Reynolds numbers in our experiments led to reach the same maximum amplitude of oscillation as some of the previous studies (A⁄ = 1), even with higher mass-damping parameter (m⁄f) in our system. The high damping in our experiments, resulted in disappearance of the lower branch in the amplitude response graphs. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Vortex-induced vibration of the cylinder has been studied extensively during the past fifty years. The study on response of an elastically mounted circular cylinder free to vibrate transverse to the flow has been started since the early work of Feng [1]. The range of problems caused by VIV has led to a large number of numerical and experimental studies on this subject, including various comprehensive review articles [2–5]. VIV was known as a destructive phenomenon and many researchers tried to reduce and diminish this harmful energy that causes failure in engineering structures. Recently, Bernitsas and his group at the University of Michigan Marine Renewable Energy Lab came up with the idea of harnessing the water current energy by utilizing vortex induced vibration [6–8]. They invented a device and called that VIVACE (Vortex Induced Vibration for Aquatic Clean Energy). This device generates clean hydrokinetic energy from slow-moving water currents by using the oscillation of the cylinder under VIV. Raghavan and Bernitsas [8], studied the Reynolds number effect on vortex induced vibration of a rigid mounted circular cylinder for Reynolds regimes right before transition from laminar to turbulent flow. They found that VIV is significantly different between the TrSL2 and TrSL3 flow regimes. ⇑ Corresponding author. E-mail address: [email protected] (A. Modir). http://dx.doi.org/10.1016/j.ijome.2016.05.001 2214-1669/Ó 2016 Elsevier Ltd. All rights reserved.
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As reported in [9], two types of response are possible for a free vibration VIV system. The first one is a small-amplitude and two branch response which corresponds to the systems with high mass-damping parameters (m⁄f). Most of these systems used air as their working fluid [10]. These branches were named as the ‘‘initial” and the ‘‘lower” branches. The second form of response is a large-amplitude, three-branch response that was believed to result from low mass-damping. Most of experiments with three branch response, used water as their working fluid [10]. The three branches were called the ‘‘initial,” ‘‘upper,” and ‘‘lower” branches. Govardhan and Williamson [11] suggested the condition A/D = 0.6, the limit to divide the regimes of three-branch and two-branch response. In Fig. 1, Feng’s minimum damping case and Khalak and Williamson’s case are compared [5]. The maximum amplitude of oscillation that can be achieved by a particular system is a subject that attracts investigators for studying different aspects of VIV. Griffin [12] for the first time collected the other investigators data and showed the relationship between maximum amplitude of oscillation for a system and a mass-damping parameter called reduced damping (SG = 2p3St2(m⁄f)). This graph is called Griffin plot and can be seen in Fig. 2. He observed that as the reduced damping increases, the related maximum amplitude of oscillation decreases, and for high mass ratio or high damping the oscillation stops. Govardhan and Williamson [13] studied the effect of Reynolds number as another effective parameter on amplitude of oscillation and changed the Griffin plot based on Reynolds number. They could find a better coverage for different experiments. Raghavan and Bernitsas [8] showed that this plot could be valid just for experiments done in the TrSL2 flow regime and their data, which was accomplished for Reynolds numbers in TrSL3 and TrBL0 regimes, was two times higher than the values obtained by Govardhan and Williamson [11]. Khalak and Williamson [9] have shown that the response of the cylinder strongly depends on the mass-damping parameter (m⁄f). They observed that for a constant value of m⁄f, by reducing mass ratio the maximum amplitude of oscillation remains almost constant, but the synchronization range increases. Govardhan and Williamson [13] studied the critical mass of a vertical cylinder in VIV. They showed that for m⁄ < 0.54, the body will oscillate over an ultimate regime of normalized velocities (U⁄) and named that, the critical mass ratio. Mass ratio plays an important role for practical applications. The goal of our research is to study the significant effect of mass ratio on vortex induced vibration of a horizontally mounted circular cylinder at high Reynolds numbers and high values of damping. 2. Physical mechanism Vortex Induced occurs when a current flows over a bluff body. For real viscous fluid flows, there will be a remarkable boundary layer on the surface of the body. For Reynolds number greater than 40, separation of the boundary layer will occur [8]. This separated layer will cause fluid rotation and this rotation causes formation of vortices, which are then shed from the back of the body and travel down the wake. During this process, vortices form and shed on the downstream side of bluff bodies in a current and induce periodic forces on the body. Fig. 3 shows this process on a flexibly mounted circular cylinder, which cause the cylinder to vibrate transverse to the flow direction. The equation of motion that normally used by many investigators to represent VIV of a circular cylinder oscillating in transverse Y direction (normal to the flow) is as below [11].
€ þ cy_ þ ky ¼ F my
ð1Þ
where m = mass of oscillating body, c = structural damping, k = spring stiffness and F = fluid force in the transverse direction. During the synchronization, a suitable approximation for force and cylinder oscillation can be given as follows:
FðtÞ ¼ F 0 sinðxt þ /Þ
Fig. 1. Amplitude response plot versus reduced velocity for two of the previous studies [5].
ð2Þ
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Fig. 2. Griffin plot which shows the maximum amplitude of oscillation versus mass-damping parameter [5].
yðtÞ ¼ y0 sinðxtÞ
ð3Þ
where x = 2pf and f = oscillation frequency. By considering that our study is basically experimental, the numerical calculations would not be considered in detail. 3. Dimensionless parameters 3.1. Reynolds number The first important parameter, and perhaps the most important parameter for studying VIV is Reynolds number (Re), which is defined by Eq. (4) (U is the free stream velocity, D is the cylinder diameter, and m is the kinematic viscosity of the fluid). Zdravkovich [14] classified the flow over a cylinder into 15 flow regimes. The range of 1 104 –2 104 < Re < 1 105–2 105, is named TrSL3 and our experiments also lie in this range. Raghavan and Bernitsas [8] compared the flow properties of TrSL2 and TrSL3 and stated that VIV behavior is significantly different in these two regimes.
Re ¼
UD
ð4Þ
m
3.2. Strouhal number The nondimensional frequency at which the vortices are shed into the wake is known as the Strouhal number (St) and is defined by Eq. (5) (fvs is the vortex shedding frequency for a stationary cylinder).
Fig. 3. Vortex induced vibration phenomenon.
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f D St ¼ v s U
ð5Þ
The relation between Reynolds number and Strouhal number for circular cylinders is shown in Fig. 4. In a wide range of Reynolds numbers, the Strouhal number varies very little, and can be considered constant. For the entire range of about 103 < Re < 105, the Strouhal number is nearly 0.2 for smooth surfaces [15], which corresponds very well to the range of Reynolds number in our experiments. Therefore, St will be taken as a constant value in any calculation. So the vortex shedding frequency (fvs) is then calculated from Eq. (6).
f vs ¼
StU D
ð6Þ
3.3. Reduced velocity (U⁄) The non-dimensional parameter used in many papers for indicating vibration amplitude, is the reduced velocity U⁄, given by Eq. (7) (fn is the natural frequency of the system in water).
U ¼
U f nD
ð7Þ
3.4. Amplitude ratio (A⁄) Normalized oscillation amplitude used in many papers, is given as A⁄ = amplitude/diameter. 4. Important parameters for the analysis of VIV 4.1. Damping ratio (f) Damping ratio is a measure for the ability of the structure to dissipate energy in a cycle of vibration. The damping of the system includes structural (frictional) damping and fluid damping. Damping ratio can be calculated by performing a free decay test, by giving an initial displacement to the cylinder and measuring the two successive peaks (yn and yn+1) of the cylinder in the water. Hence, the damping ratio can be calculated from Eq. (5).
f¼
1 y ln n 2p ynþ1
ð8Þ
4.2. Natural frequency in water Natural frequency in water is defined as bellow
1 fn ¼ 2p
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k mosc þ ma
ð9Þ
The added mass is ma = CAmd, where md = qwpD2L/4 and CA is the ideal added mass coefficient that is equal 1 for a circular cylinder [11].
Fig. 4. St–Re relationship for circular cylinders [15].
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4.3. Mass ratio (m⁄) The mass ratio is defined as the total mass of all oscillating parts plus 1/3 of the spring mass divided by the displaced fluid mass (md) [8].
m ¼
mosc mosc ¼ md qw p4 D2 L
ð10Þ
5. Lock-in Lock-in is like linear resonance in which the vibration amplitude increases as the vortex shedding frequency (fs) gets close to the natural frequency of the system (fn). The phenomenon of lock-in traditionally means that the frequency ratio (f⁄ = fosc/ fn,water) remains close to unity [10]. Accordingly, lock-in will occur when the reduced velocity (U⁄) becomes approximately equal to the inverse of the Strouhal number (1/St) which is equal to 5. In the definition given by Sarpkaya [16], during lock-in, fluid force frequency should be equal to oscillation frequency. In systems with low mass ratio, the cylinder oscillates at a frequency higher than the natural frequency of the system. The deviation of f⁄ from unity in systems under VIV is shown in [3] and [17,18]. Lock-in is a highly non-linear phenomenon, and can occur over a wide range of shedding frequencies. Lock-in also has been described as a self-limiting and self-governing phenomenon. The cylinder vibrations affect the vortex shedding procedure, and vice versa. It is known as self-limiting, due to the fact that as the cylinder displacement rises, the vortex shedding is getting weak, and accordingly decreases more motion [7]. Synchronization is defined as the matching of the vortex shedding frequency (fvs) with oscillation frequency (fosc) [18]. In many papers ‘‘Lock-in” and ‘‘Synchronization” are used interchangeably. It worth to mention that ‘‘Lock-in” and ‘‘Synchronization” have different meanings. ‘‘Lock-in” occurs when the vortex shedding frequency breaks from the Strouhal relationship and fits the natural frequency of the system. On the other side, ‘‘Synchronization” happens when vortex shedding frequency breaks from Strouhal relationship but matches a frequency that is not essentially the natural frequency of the system. Consequently, lock-in is a special case of synchronization [19]. 6. Experimental set-up Different aspects of VIV including the range of synchronization, branches of response within the synchronization, effect of mass ratio on maximum amplitude of oscillation and frequency response, are studied in this paper. Experiments are carried out in a towing-tank water channel which has 14 m length, with a cross section of 50 cm width and 60 cm depth. The purpose of the towing-tank water channel is to facilitate the study of the flow over a flexibly mounted circular cylinder in a wide range of velocities. The experimental set-up is consist of a horizontal cylinder suspended from a carriage, which rides inside the calm water channel. Three cylinders with a constant geometry but different masses are used here. All cylinders have a diameter of 6 cm and length of 38 cm with smooth surfaces. Cylinders are made from stainless steel, compressed plastic and iron, with mass ratios of 1.64, 2.26 and 3.39 respectively (Fig. 5). The weight of the overall oscillating mechanism that holds the cylinder and oscillates with it, is 1.21 kg. Aspect ratio (Length/Diameter) of all cylinders is 6.3, and the blockage ratio at every section of the channel is 11.8%. The main properties of the three different cases used in this paper are listed in Table 1. The cylinders are limited to oscillate transversely to the carriage movement by using 2 cylindrical bearings, as showing in Fig. 6. In performing the experiments, the stiffness of the springs are kept constant at 570 N/m, and the relative flow speed (the carriage speed) varied from 0.38 m/s to 1.5 m/s, yielding Reynolds number from 17,000 to 70,000. By employing an electronic inverter connected to a three-phase electromotor, the carriage speed can be changed in a broad range of velocities. The current effect is created by towing the cylinder at a constant speed inside the calm water. The relative flow passing the cylinder is equivalent with the condition that the cylinder is fixed and the water flows over it. Since tests are done in a calm water channel with sufficient time between runs, the free stream turbulence was reduced to negligible levels. The displacement of the cylinder is measured by using an advanced digital camera, recording 30 frames per second, and a ruler which allows recording the position of the cylinder during tests. 7. Results and discussion By performing the free-decay test in water, the damping ratio for three different systems obtained. Fig. 7 shows a sample of damping calculation test for an initial displacement of 40 mm. By considering that there is a metal–metal contact between bearings and the side struts, the damping associated with the bearing system is moderately high which result in high values of mass-damping parameter in the three cases reported here. Due to the facilities employed in our experiments, a wide range of reduced velocities obtained (3 < U⁄ < 11). This range is large enough to identify the lock-in region, where the higher amplitudes are observed. At lower U⁄ the response is clearly quasiperiodic, while for the slightly higher U⁄, the response becomes suddenly more periodic. Fig. 8 illustrates the time trace of a cylinder with m⁄ = 2.26 and U⁄ = 6.5 for the first four seconds.
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Fig. 5. The three different cylinders used in the present study.
Table 1 VIV model particulars. Case
mcylinder
mosc
md
(m⁄)
(f)
K (N/m)
fn
1 2 3
0.540 1.185 2.400
1.750 2.395 3.610
1.072 1.072 1.072
1.64 2.26 3.39
0.061 0.053 0.059
570 570 570
2.26 2.03 1.75
In our VIV experiments performed for three different mass ratios, maximum amplitude ratio about 1 was measured for Reynolds numbers in the range of 1.7 104 < Re < 7 104. Figs. 9 and 10 display the amplitude ratio (A/D) versus reduced velocity (U⁄) and Reynolds number (Re), for different mass ratios. Initial, Upper and lower branches are shown in Fig. 9. The amplitude of oscillation shows a decreasing trend with increase in mass ratio of the system. Based on the definition expressed in [11], it can be concluded that the upper branch disappeared for the case of m⁄ = 3.39 (A/D < 0.6). It clearly can be noticed from Fig. 10 that the amplitude of oscillation and the range of synchronization in the upper branch increase with rise in Reynolds number. As it can be observed from this figure, the amplitude of oscillation versus Reynolds number fit in one curve in the initial branch for different cylinders. It is clearly shown in Fig. 10 that by decreasing m⁄, the range of synchronization expands and shifts to the higher Reynolds numbers which shows the dependence of VIV on mass ratio. Based on the definition of Govardhan and Williamson [11], A/D = 6 is the demarcation for the three branch
Fig. 6. Main components of the VIV test model, (1) cylinder, (2) side strut, (3) linear bearings, (4) springs, (5) wheels.
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Fig. 7. A sample of damping ratio calculation, derived from the free-vibration decay test in water.
Fig. 8. A sample time-trace plot for the first four seconds of the test.
response, it was expected to observe these branches in our graphs. It can be seen by rising the reduced velocity, the amplitude ratio jumps from initial branch to the upper branch and it suddenly drops to desynchronization without experiencing the lower branch which is not a common VIV character. Raghavan and Bernitsas [8] also observed that for their high massdamping case, the lower branch disappeared. They witnessed a drop from synchronization to desynchronization without any hump in their amplitude response. It is essential to understand how the maximum amplitude of oscillation (A⁄max) changes as a function of m⁄ and f. Williamson and Govardhan [5], mentioned that the product of ‘‘m⁄f”, is the parameter which controls the maximum amplitude of oscillation. The increase in the damping can be counteracted by decreasing the mass ratio [9]. In Fig. 11 the maximum amplitude of oscillations for the three cylinders are plotted as a function of the Skop-Griffin Parameter (SG). Our results approximately fit to the classical Griffin plot, which shows the maximum amplitude of oscillation as a function of the mass-damping parameter.
7.1. Comparison In this part of paper we compare our results with three of the best-known studies on VIV of a flexibly mounted circular cylinder. In Fig. 12, the experiments done in [1,17,20], and our results are plotted in the same graph. The characteristics of these four experiments, including Reynolds number, mass ratio, mass-damping parameter ((m⁄ + CA)f) and maximum ampli-
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Fig. 9. Amplitude ratio versus reduced velocity for different mass ratios.
Fig. 10. Amplitude ratio versus Reynolds number for different mass ratios.
Fig. 11. Result of present study in the typical linear-log form of Griffin plot.
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Fig. 12. The comparison of the experiments done by Feng [1], Khalak and Williamson [17], Govardhan and Williamson [20] and the present study.
Table 2 The experimental characteristic of three well-known investigators and the present experiment. Investigator Feng [1] Khalak et al. [17] Govardhan et al. [20] Present study
Reynolds 3
4
6.0 10 –2.3 10 5.0 103–1.6 104 2.9 103–1.9 104 1.2 104–6.8 104
m⁄
(m⁄ + CA)f
A/D
250 2.4 1.19 1.64
0.36 0.013 0.011 0.16
0.55 0.94 1.0 1.0
Fig. 13. Nondimentional frequency of response, f⁄, versus nondimentional velocity, U⁄.
tude of oscillation are listed in Table 2. As stated above, added mass coefficient (CA) is assumed to have a value of unity for a circular cylinder [13]. In Fig. 12, we may compare our amplitude response plot for m⁄ = 1.64 and f = 0.061 with the response measured by Feng [1], in whose experiments m⁄ = 248 and f = 0.0014. Since the present results have lower mass-damping than those of Feng, (less than half of his value), we find much greater amplitudes, with nearly twice the maximum amplitude of what Feng could achieve. As one can observe in Fig. 12, the range of synchronization and the maximum amplitude of oscillation in our experiments are very close to the one measured by Khalak and Williamson [17], while the mass-damping parameter in our experiments is 12 times higher. One of the reasons for accomplishing the same maximum amplitude in these two experiments can be explained due to the higher Reynolds numbers and consequently higher lift coefficient [8] in our case.
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Fig. 14. Oscillation frequency of cylinders versus velocity.
By comparing the results of [17] with [20], it can be concluded that by keeping constant the mass ratio parameter ((m⁄ + CA)f) for almost the same Reynolds number, maximum amplitude of oscillation stays constant but the synchronization range increases by decreasing the mass ratio of the system. This explanation can be an evidence of disappearance of lower branch in our experiment. 7.2. Frequency response The frequency responses of the three cylinders are shown in Fig. 13. In this figure, the ratio of the oscillation frequency to the natural frequency in water (f⁄ = fosc/fn,water) is plotted versus reduced velocity. The Strouhal frequency (inclined line) is also shown for comparison. In the initial branch of oscillation, cylinders oscillate at lower frequency than the natural frequency of the system and by increasing the reduced velocity throughout the upper branch, the nondimentional frequency deviates slowly from unity, reaching its final value near 1.3 for U⁄ = 11. This does not correspond to either Strouhal frequency of a stationary cylinder (f⁄ = 0.2U⁄) or the natural frequency of the system (f⁄ = 1), but lies between them. At reduced velocities close to 7, the cylinders oscillate at the natural frequency of the system which is the definition of lock-in. By referring to Fig. 9, it can be observed that cylinders oscillate at the maximum amplitude of oscillation in this value of reduced velocity. The frequency of oscillation versus current velocity (U) for all cylinders are shown in Fig. 14. It is interesting to mention that regardless the mass ratio of the cylinders, all of them oscillate at the same frequency for a specific fluid velocity and the frequency response curves collapse into one curve in the synchronization regime. By increasing fluid velocity, it can be observed that the oscillation frequency follows an increasing trend for all cylinders. 8. Conclusions The effect of mass ratio on vortex-induced vibration of an elastically mounted rigid circular cylinder was tested in a towing-tank water channel for three different cylinders with the same geometry but different masses. The mass ratio of the cylinders were m⁄ = 1.64, 2.26 and 3.39, while the damping ratio and spring stiffness were kept constant. The experiments were done in a wide range of Reynolds numbers (1.7 104 < Re < 7 104) in a towing-tank water channel. All our results for high damping systems show strong depends of VIV on mass ratio. The maximum amplitude of oscillation obtained during the experiments was considerably close to the results obtained in [17], while our mass-damping parameter was noticeably higher. Different amplitude response branches were clearly shown in different plots and the following conclusions can be made: (1) The maximum amplitude of oscillation and the range of synchronization strongly depend on mass ratio. (2) As m⁄ decreases, the maximum amplitude of oscillation increases and the range of synchronization expands and shifts to higher Reynolds numbers. (3) The range of synchronization in upper branch increases with rise in Reynolds number. (4) The upper branch disappeared (A⁄ < 0.6) for the highest mass ratio case (m⁄ = 3.39). (5) In our experiments with high damping parameter, the amplitude of oscillation in upper branch fell sharply to the desynchronization without experiencing a detectable lower branch.
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(6) The nondimentional frequency stays close to unity in upper branch and deviates slowly from 1 with increasing the reduced velocity. In lower branches, the nondimensional frequency of oscillation was greater than unity, but lower than the Strouhal frequency. (7) All three cylinders oscillated at the same frequency for different fluid velocities. (8) By comparing our results with [17,20], it can be concluded that maximum amplitude of oscillation principally dependents on the mass-damping parameter (m⁄f), whereas the regime of synchronization (measured by the range of U⁄), for a given value of m⁄f, is dependent primarily on the mass ratio, only.
References [1] C.C. Feng, The Measurement of Vortex-Induced Effects in Flow Past a Stationary and Oscillating Circular Cylinder and d-Section Cylinders (Master’s thesis), University of British Columbia, Vancouver, Canada, 1968. [2] T. Sarpkaya, Vortex-induced oscillations, J. Appl. Mech. 46 (1979) 241–258. [3] P.W. Bearman, Vortex shedding from oscillating bluff bodies, Annu. Rev. Fluid Mech. 16 (1984) 195–222. [4] C.H.K. Williamson, Vortex dynamics in the cylinder wake, Annu. Rev. Fluid Mech. 28 (1996) 477–539. [5] C.H.K. Williamson, R. Govardhan, Vortex-induced vibrations, Annu. Rev. Fluid Mech. 36 (2004) 413–455. [6] M.M. Bernitsas, K. Raghavan, Y. Ben-Simon, E.M. Garcia, VIVACE (Vortex Induced Vibration Aquatic Clean Energy): a new concept in generation of clean and renewable energy from fluid flow, J. Offshore Mech. Arct. Eng. ASME Trans. 130 (4) (2008) 041101–041115. [7] M.M. Bernitsas, Y. Ben-Simon, K. Raghavan, E.M.H. Garcia, The VIVACE converter: model tests at high damping and Reynolds Number around 105, J. Offshore Mech. Arct. Eng. Trans. ASME 131 (2009) 1. [8] K. Raghavan, M.M. Bernitsas, Experimental investigation of Reynolds number effect on vortex induced vibration of rigid cylinder on elastic supports, Ocean Eng. 38 (2011) 719–731. [9] A. Khalak, C.H.K. Williamson, Motions forces and mode transitions in vortex induced vibrations at low mass-damping, J. Fluids Struct. 13 (1999) 813– 851. [10] J.T. Klamo, Effects of Damping and Reynolds Number on Vortex-Induced Vibrations (Ph.D. thesis), California institute of Technology, 2007. [11] R. Govardhan, C.H.K. Williamson, Defining the ‘modified Griffin plot’ in vortex-induced vibration: revealing the effect of Reynolds number using controlled damping, J. Fluid Mech. 561 (2006) 147–180. [12] O.M. Griffin, R.A. Skop, S.E. Ramberg, The resonant, vortex-excited vibrations of structures and cable systems, in: Annual Offshore Technology Conference, May, Houston, TX, 1975. [13] R. Govardhan, C.H.K. Williamson, Critical mass in vortex-induced vibration of a cylinder, Eur. J. Mech. Fluids 23 (2004) 17–27. [14] M.M. Zdravkovich, Flow Around Circular Cylinders Volume 1: Fundamen-tals, Oxford Science Publications, 1997. [15] R.D. Blevins, Flow-Induced Vibration, second ed., Van Nostrand Reinhold, New York, 1990. [16] T. Sarpkaya, Fluid forces on oscillating cylinders, J. Waterw. Port Coastal Ocean Div. 104 (1978) 275–290. [17] A. Khalak, C.H.K. Williamson, Fluid forces and dynamics of a hydroelastic structure with very low mass and damping, J. Fluids Struct. 11 (1997) 973– 982. [18] A. Khalak, C.H.K. Williamson, Investigation of relative effects of mass and damping in vortex-induced vibration of a circular cylinder, J. Wind Eng. Ind. Aerodyn. 69 (1997) 341–350. [19] J.T. Klamo, A. Leonard, A. Roshko, The effects of damping on the amplitude and frequency response of a freely vibrating cylinder in cross-flow, J. Fluids Struct. 22 (2006) 845–856. [20] R. Govardhan, C.H.K. Williamson, Modes of vortex formation and frequency response of a freely vibrating cylinder, J. Fluid Mech. 420 (2000) 81–930.