- Email: [email protected]
Fluid excitation of an oscillating circular cylinder in cross-flow
European Journal of Mechanics B/Fluids 29 (2010) 364–368
Contents lists available at ScienceDirect
European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
Fluid excitation of an oscillating circular cylinder in cross-flow A. Barrero-Gil ∗ , P. Fernandez-Arroyo ETSIA Aeronáuticos, Instituto Universitario ‘Ignacio Da Riva’, Universidad Politecnica de Madrid, E-28040 Madrid, Spain
article
info
Article history: Received 30 July 2009 Received in revised form 21 December 2009 Accepted 23 March 2010 Available online 31 March 2010 Keywords: Vortex-induced-vibration Circular cylinder Fluid forces Least-squares
abstract This paper presents an experimental technique developed to identify the fluid excitation force of a circular cylinder during Vortex-Induced Vibrations (VIV). To this end, an actuator is used to provide controlled damping and a Van der Pol model is proposed to describe the fluid excitation force, where the associated parameters are identified from experiments by a least square fitting. The Reynolds number was in the range 1650–4950, and the technique is validated with experimental free vibration data of an elastically supported circular cylinder. Also, comparison with free vibration measurements from other experiments carried out with similar Reynolds numbers is presented. © 2010 Elsevier Masson SAS. All rights reserved.
1. Introduction Vortex-Induced Vibrations (VIV) constitutes an important problem in the aero/hydro-elasticity field for several reasons. From a practical side, it is well known that VIV can induce severe vibrations in civil-engineering structures, such us slender chimney stacks, bridges, tall buildings, electric power lines and heat exchange tubes, as well as in marine structures like offshore risers or marine pipelines. From a scientific point of view, VIV constitutes a challenging problem, where there is an interesting feedback between the vibrating body and the flow field around it. Basically, VIV is due to the periodic shedding of vortices that can excite an elastic body into resonant oscillations when the vortex shedding frequency and body frequency are sufficiently close one to another. Under certain conditions significant oscillations can emerge (principally, normal to the incident flow). Then, a coupling between the oscillating body and the flow field around it develops with two relevant consequences: (i) there is a range of flow velocities where vortex shedding frequency is synchronized with the frequency of oscillation (lock-in regime), and (ii) the cylinder response may exhibit hysteresis, with jumps in oscillation amplitude and in the fluid forces acting on the body. Because it is of practical and scientific interest, since the early 1960’s a large number of fundamental studies of VIV have been carried out, many of which are reviewed by, among others, Bearman [1], Sarpkaya [2], and Williamson and Govardhan [3]. The experimental study of VIV has been mainly based on two different approaches. One approach, in an attempt to
∗
Corresponding author. E-mail address: [email protected] (A. Barrero-Gil).
0997-7546/$ – see front matter © 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2010.03.005
simulate the flow-induced vibration directly, consists of studying experimentally the canonical problem of an elastically supported rigid body (normally a circular cylinder) restricted to move in the transverse direction y, as sketched in Fig. 1. This problem can be described by the following structural parameters: a cylinder with mass per unit length m, diameter D, length L, a characteristic frequency of oscillation fN (normally defined in still air), and mechanical damping ratio ζ . The fluid parameters are the velocity of the incoming flow U (generally turbulence is not considered) and fluid properties ρ (density) and ν (kinematic viscosity). Therefore, taking ρ , D, and fN as the fundamental physical quantities, the normalized amplitude A∗ = A/D, and frequency of oscillation f ∗ = f /fN presents the following dependence: A∗ , f ∗ = Φ (m∗ , ζ , U ∗ , Re, L/D),
(1)
where m∗ = m/(ρ D2 ) is the mass ratio (the dimensionless number typifying the ratio of the mean density of the body to the density of the flow), U ∗ = U /(fN D) is the reduced velocity, and Re = UD/ν stands for the Reynolds number. Over the years, extensive research has been conducted for the experimental determination of this relationship (Eq. (1)) (for example, see [4] for m∗ 1 and [5] for m∗ ∼ 1). The other approach is focused on the measurement of the fluid forces on an harmonically forced body (length L, cross-section side D) with amplitude A and frequency f in a uniform flow. Again, circular cylinders have been studied extensively [6–8]. In this case the lift force is usually represented by a force coefficient CY and a phase angle φ with the force leads the displacements, with the following dependence, CY , φ = Φ (A∗ , U ∗ , Re, L/D).
(2)
A. Barrero-Gil, P. Fernandez-Arroyo / European Journal of Mechanics B/Fluids 29 (2010) 364–368
The fluid forces acting on the body are usually measured by means of pressure taps located appropriately on the surface of the model or by means of piezoelectric gauges. The former method requires a large number of pressure taps for accuracy and the latter method requires that the inertial force FI (my¨ , being m and y¨ , respectively, the mass and acceleration of the body) must be subtracted from the total force registered by the piezoelectric gauge to obtain the fluid force. As the fluid forces FF are of order ρ U 2 D, m∗ must be of order unity for accuracy in the measurements [note that FF /FI ∼ U ∗ /(2π m∗ A∗ )]. That is the reason why these measurements are usually carried out in water facilities. However, another approach is possible when the fluid forces are determined from their effect on the motion of the mechanical system. In this case, the situation is more representative of the real fluid-structure interaction problem. However, to the author’s knowledge, this approximation has not been followed much by researchers of VIV. This is the strategy followed in this study. In the method proposed here a mathematical form with two undetermined parameters for the fluid excitation force is assumed. As will be shown later, they will be a function of the reduced velocity U ∗ and a least squares method will help us to identify these parameters from a designed experimental procedure. Fluid excitation means the fluid force component associated with the body velocity. For high mass ratio, the fluid force component in phase with cylinder acceleration (the added mass) has no practical importance in the VIV dynamics. This was clearly shown for the first time by Bearman [1]. Let us summarize his reasoning: a system such us that shown in Fig. 1 has a governing equation of motion given by 1
m(¨y + 4π ζ fN y˙ + 4π 2 fN2 y) =
ρ U 2 DCY (t ), (3) 2 where CY is the fluid force coefficient in the transverse direction. In the lock-in region (or resonance range), a good approximation of the fluid force and the cylinder displacement can be described as CY (t ) = CY sin(2π ft + φ),
(4)
y(t ) = A sin(2π ft ),
where f is the oscillation frequency and φ is the phase angle between the fluid force and the cylinder displacement. Substituting Eq. (4) in Eq. (3) and equating sine and cosine terms, the solution for the response amplitude and frequency is ∗
A =
f
∗
CY sin φ 16π 2 m∗ ζ
= 1−
U ∗2
f∗
CY cos φ U ∗2 8π 2 m∗ A∗
,
1/2
(5)
.
For a large mass ratio, m∗ 1, from Eq. (5) one can see that f ∗ −→ 1 (U ∗2 /8π 2 ∼ 1 and experiments show that CY 0 cos φ ∼ 1), and (U ∗ /f ∗ ) −→ St−1 (herein St is the stationary Strouhal number). Thus, one may deduce that response amplitude is a function of reduced velocity, mass-damping parameter (m∗ ζ ) and fluid force in phase with cylinder velocity (fluid excitation, CY sin φ ). Note that m∗ 1 is the situation in wind engineering and for most civilengineering structures, where air is the fluid that surrounds the body. In the following section, the experimental apparatus used to identify fluid excitation is described, as well as the experimental procedure. In Section 3, experimental results are presented. We show contours of fluid excitation as a function of the governing parameters (A∗ and U ∗ ), and a comparison with some previous experimental results is presented. Also, free vibrations experiments have been carried out to compare with the analytical prediction of the body response using the previously identified fluid excitation data. Finally, relevant results are summarized in Section 4.
365
Fig. 1. Flow-structure parameters in the VIV canonical problem.
2. Experimental details 2.1. Experimental set-up and procedure The work which is reported here has a main purpose: to measure fluid forces of an oscillating cylinder under cross flow by means of an inverse method. To this end, a spring-mass-dashpot system with an electromagnetic device to provide additional excitation has been designed. As can be seen in Fig. 2a, a circular cylinder of length L = 33 cm was supported elastically by means of two aluminium beams (40 cm long). The cylinder has a smooth surface finishing and is made out of plastic, and it was under the action of an airstream supplied by a small open-test section wind tunnel of rectangular area of 25 cm height, 40 cm width. The cylinder was situated 6 cm downstream of the test section entrance. In all cases, the width of the cylinder (D) was 3.3 cm, giving an aspect ratio (L/D) of 10, and the wind speed U was varied in the range 50–150 cm/s (so that, the Reynolds number, defined as Re = UD/ν , was varied between 1650 and 4950). The free-stream turbulence level on the centerline was below 2%. The transverse vibration, y(t ), was measured with a non-contact laser vibrometer (with a resolution of 60 µm), and the wind tunnel speed was measured with a hot wire anemometer (0.01 m/s of resolution). To provide accurate control in damping, a voice coil was used as actuator. The voice coil produces a force that is proportional to the current drive through the coil. If Va (t ) is the voltage across the coil, R it’s electrical resistance and x(t ) the displacement of the coil, then the force Fa supplied by the voice coil is given by (the coil inductance and capacitance can be neglected), Fa (t ) =
K R
Va (t ) −
K2 R
+ b x˙ ,
(6)
where K is a constant and b another one which takes into account the eddy current effects. If Va (t ) is proportional to the transverse velocity of vibration, y˙ (t ), and x(t ) is proportional to the transverse vibration amplitude of the square cylinder, then Fa (t ) = C y˙ (t ),
(7)
and therefore the voice coil introduces an external damping term in the system dynamics. As can be seen in the block diagram shown in Fig. 2b, after the elastic system block the output voltage of the laser vibrometer (proportional to y) is differentiated by means of a passive highpass filter with a cut-off frequency of 194 Hz (as the natural
366
A. Barrero-Gil, P. Fernandez-Arroyo / European Journal of Mechanics B/Fluids 29 (2010) 364–368
a
b Fig. 2. Experimental set-up (a) and schematic block diagram (b).
frequency of the system was 4.7 Hz the phase lag introduced is less than 2◦ ), so that V2 (t ) = RF CF (dVacc (t )/dt ) = k1 y˙ (t ) [RF and CF are, respectively, the resistance and capacity of the filter, k1 is a constant]. Then, V2 is amplified and sent to the power amplifier that drives the voice coil with a voltage proportional to the velocity of oscillation, Va (t ) = k1 k2 y˙ (t ) (and hence Fa = C y˙ ). The magnitude of the damping introduced, C , is varied with a potentiometer and its sign can be changed if the polarity of Va is reversed, providing accurate control of positive and negative damping. At the same time, the laser vibrometer signal is recorded by a PC at a rate of 500 Hz. To account for nonlinearities in the mechanical system (due principally to mechanical and fluid damping in still air) Eq. (3) is modified introducing a nonlinear term of the Van der Pol type (i.e. we assume that the damping force in the system can be represented as the sum of a linear damping term and a cubic damping term). Then, the dynamics of the system (see Fig. 2a) can be represented by the following equation,
tunnel is switched on at the desired velocity the voice coil is switch on and the parameters describing fluid excitation force, b1 (U ∗ ) and b2 (U ∗ ), are determined from the response by a least square fitting. For a particular velocity the test was repeated three times and the mean values were selected. The range of reduced velocity covered during the experiments was that within which VIV usually takes place, 3.5 < U ∗ < 10. Fluid excitation force was identified for eighteen values of reduced velocity. In all cases, the oscillation was registered over a 5 minute period (over 1400 cycles of oscillations). As shown before, the mechanical damping and fluid forces are smaller than the inertial and stiffness forces, and experiment shows that in the resonance range the response is essentially a sinusoidal function. Out of the resonance range, fluid forcing exhibits somewhat non-sinusoidal behaviour. However, as in previous studies [7,8], only the fluid excitation component (CY sin φ ) that occurs at the body oscillation frequency will be presented here. It is computed by means of a Fourier analysis of the identified fluid force.
m(¨y + 4π 2 fn2 y) + a1 y˙ + a2 y2 y˙ = FY (t ),
2.2. Post-processing and identification procedure
(8)
which represents a balance between the linear inertial and stiffness forces, the nonlinear mechanical damping force, and the fluid excitation force FY . Herein m is the equivalent mass per unit length of the system (mass of the cylinder plus the modal mass of the beam system), and fN is the natural frequency of the oscillations. First of all, a calibration of the installation was done. The stiffness k of the installation was measured by adding calibrated weights at both ends of the body and measuring its vertical displacement. Later, with the voice coil switch off, the body was excited with an initial displacement giving a frequency of oscillation of 4.7 Hz [and hence the equivalent mass per unit length of the system was deduced as m = k/(2π fn )2 ]. Finally, in still air (U ∗ = 0), the voice coil was switched on providing excitation (that is negative damping) for low amplitudes of vibration. The amplitude of oscillations grew until a limit cycle of oscillations around A∗ = 1 was reached. From the measured oscillations parameters a1 and a2 were computed by a least square fitting. The test was carried out three times and the mean values were selected. The repeatability of the procedure was fairly good, with a standard deviation of 0.018 (in the case of a1 ) and 0.028 (a2 ). To indirectly measure the fluid excitation force, FY (t ) is assumed to be described again by a Van der Pol term [9]. The rationale behind such a model is related to the experimental evidence that oscillations due to VIV have a self-limiting character, as were the electrical oscillations in the circuits studied by Van der Pol [10]. So that, FY (t ) = b1 y˙ + b2 y2 y˙ , where b1 (U ∗ ) and b2 (U ∗ ) are functions of the flow velocity (reduced velocity). After the wind
Putting Eq. (8) in a dimensionless form (η = y/D, τ = 2π ft) and taking into account that FY (t ) = b1 y˙ + b2 y2 y˙ , one obtains
η00 = −Ω η + k1 η0 + k2 η2 η0 ,
(9)
where Ω = (fn /f ) , k1 (U ) = (b1 (U ) − a1 )/(2π fmD), and k2 (U ∗ ) = (b2 (U ∗ ) − a1 )D/(2π fm). To find Ω (U ∗ ), k1 (U ∗ ), and k2 (U ∗ ) a least square fit has been used from the measured displacements (y, or in dimensionless form η) at the sampled points. At any instant Eq. (9) needs to be satisfied so that it can be expressed in a matrix form, 2
∗
∗
−→→ − →00 0− → η = [− η η2 η0 − η ][k1 k2 − Ω ]T = Aθ = b, (10) − → where θ = (k1 k2 Ω )T and b = η00 . Such a redundant system can be solved, in a least square sense, doing
θ = (AT A)−1 AT b.
(11)
Numerical differentiation was carried out using a centered five point method. For example, the velocity of oscillation, η0 , was computed as η0 (ti ) = [−η(ti+2 )+ 8η(ti−1 )− 8η(ti−1 )−η(ti−2 )]/(12∆); ∆ = ti − ti−1 being the sampling time. Next, a numeric filter with a cut-off frequency of fc /fn = 5 was applied to eliminate high frequency noise introduced by numerical differentiation. It should be noted that as it is necessary to compute the velocity and the acceleration of the cylinder carefully, the frequency of sampling must be as high as possible. In the experiments data were recorded at a rate of 500 Hz. Also, for better identification it is desirable that the
A. Barrero-Gil, P. Fernandez-Arroyo / European Journal of Mechanics B/Fluids 29 (2010) 364–368
367
Table 1 Comparison of the identified fluid excitation with the fluid excitation measured by Morse and Williamson [8] and Hover et al. [7] in forced oscillation tests. Comparison is only possible when the experimental conditions are similar. [8]: Re = 4000, L/D = 10. [7]: Re = 3800, L/D = 19.2. A∗
Morse and Williamson [8]
Present data
Re
U∗
CY sin φ
Re
U∗
CY sin φ
0.55 0.65 0.75 0.9
4000 4000 4000 4000
7.87 7.87 7.87 7.87
0.1 −0.2 −0.4 −0.6
4000 4000 4000 4000
7.87 7.87 7.87 7.87
−0.25 −0.45 −0.8
A∗
Hover et al. [7]
0.5 0.65 0.9 1
0.03
Present data
Re
U∗
CY sin φ
Re
U∗
CY sin φ
3800 3800 3800 3800
7.43 7.43 7.43 7.43
0
3800 3800 3800 3800
7.43 7.43 7.43 7.43
−0.26 −0.65 −0.9
−0.2 −0.4 −0.6
0.1
Table 2 Estimated values for different noise levels (or relative error). True values
Estimated values
= 0.01 k1 = 0.01 k2 = −0.045
= 0.05
= 0.1
0.0098
0.0097
0.0090
−0.0438
−0.0413
−0.0386
experimental setup presents low values of a1 (<0 and controlled by the actuator) and a2 (i.e. that the mechanical system presents a small nonlinearity). 2.3. Experimental uncertainties Let us now to discuss the main sources of uncertainties in our measurements. The laser displacement used (M7 from MEL Mikroelektronik) had a resolution of ±0.06 mm, and therefore if the oscillation amplitude of the cylinder was higher than A∗ = 0.05 the measured displacement is known to within an error of less than 3.6%. The resolution of the hot wire was ±0.01 m/s and during experiments the mean flow velocity measured was known within an error of the order of 1%. On the other hand, the diameter of the cylinder was measured with a vernier caliper within an error of 0.25% and the weights added to calculate the stiffness of the mechanical system were measured with a balance to a resolution of ±0.1 g (relative error always less than 2%). Then, taking into account the error propagation, the normalized velocity A∗ was known within a nominal error of the order of 4% and the reduced velocity within a nominal error of 3.5%. The mode in which uncertainties in the measurement of the displacement are propagated in the velocity and acceleration computations (and hence in the identified parameters) is not easy to estimate. To test the influence of uncertainties in the estimated parameters a sequence of white noise was added to simulated data. These data were obtained numerically solving Eq. (9) with a fourth order Runge–Kutta algorithm. Table 2 shows some typical values for various noise levels. It seems that the estimation is sensitive to the influence of noise and coefficients are poorly estimated for high noise. 3. Results 3.1. Fluid excitation The identified fluid excitation coefficient is shown in Fig. 3 in the parameter space (U ∗ , A∗ ). With comparison purposes values from the experiments carried out by Morse and Williamson and Hover et al. are also presented in Table 1. The contour of zero fluid excitation (CY sin φ = 0) is particularly interesting. It divides the region in the (U ∗ A∗ ) plane where energy transfer is from the flow to the body (CY sin φ > 0; excitation)
Fig. 3. Contour plot of the identified fluid excitation, CY sin φ . Contour interval is 0.2. Note that x axis stands for the normalized velocity as well as the Reynolds number (upper one). There is a clear correspondence, as Re = U ∗ (fN D2 /ν), where β = fN D2 /ν is the Stokes number (=511 in this case). Also, three distinct zones (corresponding to initial, upper,and lower branch) are identified at the top of the figure.
from that where the energy transfer is from the body to the flow (CY sin φ < 0). With respect to its shape, we can distinguish four different zones or, following the nomenclature introduced by Khalak and Williamson [5], branches. The initial branch (3.5 < U ∗ < 5), where fluid excitation increases with reduced velocity, is followed by the upper branch (5 < U ∗ < 6.5) where some hysteretic behavior can be seen, as fluid excitation is not univocally determined by normalized amplitude (for example, for U ∗ = 5.5 three possible values of CY sin φ can be distinguished). For (6.5 < U ∗ < 9) fluid excitation is fairly constant. In the literature this is known as the lower branch. Finally, for U ∗ < 9 fluid excitation drops rapidly. On the other hand, the zero fluid excitation boundary represents the maximum amplitude of oscillation achievable during VIV (when the mass damping parameter is very low). In this case at a Reynolds number of 2933 (U ∗ = 5.74) and m∗ ζ −→ 0, the maximum amplitude of oscillation would be of 0.81. The value proposed by Govardham and Williamson [11] for the above mentioned Reynolds number is 0.85, close to the one proposed here. 3.2. Free vibration experiments To test the identified fluid excitation force, the same experimental arrangement without the action of the voice coil was used to look for response amplitude versus reduced velocity. The experimental protocol was always the same: fixing the airstream velocity, the cylinder was exposed over a sufficiently long time until a steady state of oscillation was reached; then, the oscillation was registered during 60 s. Next, the flow velocity was increased by a small amount with the cylinder still oscillating and the operation was repeated. Experimental results are compared to analytical predictions with the identified values of fluid excitation, as shown in Fig. 4. From the figure it can be seen that the maximum amplitude of oscillations is well captured, as well as the reduced velocity which occurs. Also, the range of reduced velocities where oscillations are of significance is well captured. It should be noted that analytical values can be obtained solving iteratively Eq. (5): for a fixed reduced velocity, the normalized
368
A. Barrero-Gil, P. Fernandez-Arroyo / European Journal of Mechanics B/Fluids 29 (2010) 364–368
knowledge of the fluid excitation component of the force could be enough to describe VIV until values of the mass ratio as low as 10. 4. Conclusions
Fig. 4. Comparison of measured free vibration (black circles) and predicted with the identified fluid excitation force (dashed line). m∗ = 260, ζ = 0.0024.
The determination of fluid forces for an oscillating circular cylinder in cross flow is a problem of engineering significance, as its VIV response can be estimated with the aid of measured fluid forces. In this paper, an experimental procedure to identify fluid excitation on a spring-mounted circular cylinder in cross flow has been presented. While we can control linear damping on the experimental arrangement, an accurate identification of fluid excitation is possible. Contrary to the traditional employed case of forced oscillations, fluid excitation is identified from a fluidstructure interaction experimental set-up, creating a more realistic approach to the problem. We have assumed a nonlinear form for fluid excitation and identified it for a circular cylinder of aspect ratio of 10 in the reduced velocity range of 3.5 < U ∗ < 10 (1790 < Re < 5110) and normalized amplitude 0 < A∗ < 1. Results show good correspondence with those obtained from previous forced oscillations experiments. Also, the identified fluid excitation has predicted well the free vibrations of the same elastically supported circular cylinder, and comparison with free vibrations for other experimental studies is good. Finally, it must be noted that the technique here described can be used to identify fluid excitation (and hence VIV response) on other geometries prone to VIV with high interest from the civil engineering point of view like, among others, square or rectangular prisms and H-section prims. In these cases, as the separation of the flow generally occurs at fixed points, the dependence upon the Reynolds number is expected to be weaker that for the case of a circular cylinder. Acknowledgements The authors wish to thank the referees for their valuable comments. References
Fig. 5. Comparison of free vibration data from [12] (open circles) with the predicted values using the identified fluid excitation force (solid line).
amplitude is increased from zero (and thus the fluid excitation coefficient evaluated) until a value where Eq. (5) is satisfied within a prescribed tolerance. 3.3. Prediction for moderate m∗ It was shown in the introduction that fluid excitation (CY sin φ ) very approximately governs the dynamics of VIV for high mass ratio values. However, Fig. 5 shows experimental results of the A∗ vs. U ∗ behaviour from [12] for m∗ = 7.1, ζ = 0.0014, β = fn D2 /ν = 520, together with analytical predictions with our identified values of fluid excitation (where β = 511). As can be seen, the agreement is not so bad. The peak amplitude predicted is close to the one measured by Klamo and the range of resonance is similar in both cases. We think that this is a somewhat expected result. Maull and Kaye [13], in their experiments where m∗ ' 12 found only a very little shift of resonances of the response of a flexible cylinder in waves (f ∗ was very close to one). On the other hand, the range of resonance predicted by the fitting formula introduced by Govardham and Williamson [14] is quite constant for m∗ > 10. Thus, one might expect that fluid excitation governs approximately the A∗ vs. U ∗ behavior until m∗ ' 10. Then, the
[1] P. Bearman, Vortex shedding from oscillating bluff bodies, Ann. Rev. Fluid Mech. 16 (195) (1984) 222. [2] T. Sarpkaya, A critical review of the intrinsic nature of vortex-induced vibrations, J. Fluids Struct. 19 (389) (2004) 447. [3] C.H. Williamson, R. Govardhan, Vortex-induced vibrations, Ann. Rev. Fluid Mech. 36 (413) (2004) 455. [4] C.C. Feng, The measurement of vortex induced effects in flow past a stationary and oscillating circular and D-section cylinder, M.A.Sc. Thesis, University of British columbia, Canada, 1968. [5] A. Khalak, C.H. Williamson, Dynamics of a hydroelastic cylinder with very low mass and damping, J. Fluids Struct. 10 (1996) 455–472. [6] T. Staubli, Investigation of oscillating forces on a vibrationg cylinder in cross flow, Doctoral dissertation, ETH Zurich, 1983. [7] F.S. Hover, A.H. Techet, M.S. Triantafyllou, Forces on oscillating uniform and tapered cylinders in cross flow, J. Fluid Mech. 363 (1998) 97–114. [8] T.L. Morse, C.H. Williamson, Fluid forcing, wake modes, and transitions for a cylinder undergoing controlled oscillations, J. Fluids Struct. (2009) doi:10.1016/j.jfluidstructs.2008.12.003. [9] E. Simiu, H. Scanlan, Wind Effects on Structures, Wiley, New York, 1996. [10] R.D. Gabbai, H. Benaroya, An overview of vortex-induced vibration of circular cylinders, J. Sound Vibration 282 (2005) 575–616. [11] R. Govardham, C.H. Williamson, Defining the ‘modified Griffin plot’ in vortexinduced vibration: revealing the effect of Reynolds number using controlled damping, J. Fluid Mech. 561 (2006) 147–180. [12] J.T. Klamo, Effects of damping and Reynolds number on vortex-induced vibrations, Ph.D. Thesis, Caltech, 2007. [13] D.J. Maull, D. Kaye, Oscillations of a flexible cylinder in waves, in: Proc. 5th Conf. on Behaviour of Offshore Structures, BOSS, Trondheim, vol. 2, 1988, pp. 535–547. [14] R. Govardham, C.H. Williamson, Modes of vortex formation and frequency response from a freely-vibrating cylinder, J. Fluid Mech. 420 (2000) 85–130.
Contents lists available at ScienceDirect
European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
Fluid excitation of an oscillating circular cylinder in cross-flow A. Barrero-Gil ∗ , P. Fernandez-Arroyo ETSIA Aeronáuticos, Instituto Universitario ‘Ignacio Da Riva’, Universidad Politecnica de Madrid, E-28040 Madrid, Spain
article
info
Article history: Received 30 July 2009 Received in revised form 21 December 2009 Accepted 23 March 2010 Available online 31 March 2010 Keywords: Vortex-induced-vibration Circular cylinder Fluid forces Least-squares
abstract This paper presents an experimental technique developed to identify the fluid excitation force of a circular cylinder during Vortex-Induced Vibrations (VIV). To this end, an actuator is used to provide controlled damping and a Van der Pol model is proposed to describe the fluid excitation force, where the associated parameters are identified from experiments by a least square fitting. The Reynolds number was in the range 1650–4950, and the technique is validated with experimental free vibration data of an elastically supported circular cylinder. Also, comparison with free vibration measurements from other experiments carried out with similar Reynolds numbers is presented. © 2010 Elsevier Masson SAS. All rights reserved.
1. Introduction Vortex-Induced Vibrations (VIV) constitutes an important problem in the aero/hydro-elasticity field for several reasons. From a practical side, it is well known that VIV can induce severe vibrations in civil-engineering structures, such us slender chimney stacks, bridges, tall buildings, electric power lines and heat exchange tubes, as well as in marine structures like offshore risers or marine pipelines. From a scientific point of view, VIV constitutes a challenging problem, where there is an interesting feedback between the vibrating body and the flow field around it. Basically, VIV is due to the periodic shedding of vortices that can excite an elastic body into resonant oscillations when the vortex shedding frequency and body frequency are sufficiently close one to another. Under certain conditions significant oscillations can emerge (principally, normal to the incident flow). Then, a coupling between the oscillating body and the flow field around it develops with two relevant consequences: (i) there is a range of flow velocities where vortex shedding frequency is synchronized with the frequency of oscillation (lock-in regime), and (ii) the cylinder response may exhibit hysteresis, with jumps in oscillation amplitude and in the fluid forces acting on the body. Because it is of practical and scientific interest, since the early 1960’s a large number of fundamental studies of VIV have been carried out, many of which are reviewed by, among others, Bearman [1], Sarpkaya [2], and Williamson and Govardhan [3]. The experimental study of VIV has been mainly based on two different approaches. One approach, in an attempt to
∗
Corresponding author. E-mail address: [email protected] (A. Barrero-Gil).
0997-7546/$ – see front matter © 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2010.03.005
simulate the flow-induced vibration directly, consists of studying experimentally the canonical problem of an elastically supported rigid body (normally a circular cylinder) restricted to move in the transverse direction y, as sketched in Fig. 1. This problem can be described by the following structural parameters: a cylinder with mass per unit length m, diameter D, length L, a characteristic frequency of oscillation fN (normally defined in still air), and mechanical damping ratio ζ . The fluid parameters are the velocity of the incoming flow U (generally turbulence is not considered) and fluid properties ρ (density) and ν (kinematic viscosity). Therefore, taking ρ , D, and fN as the fundamental physical quantities, the normalized amplitude A∗ = A/D, and frequency of oscillation f ∗ = f /fN presents the following dependence: A∗ , f ∗ = Φ (m∗ , ζ , U ∗ , Re, L/D),
(1)
where m∗ = m/(ρ D2 ) is the mass ratio (the dimensionless number typifying the ratio of the mean density of the body to the density of the flow), U ∗ = U /(fN D) is the reduced velocity, and Re = UD/ν stands for the Reynolds number. Over the years, extensive research has been conducted for the experimental determination of this relationship (Eq. (1)) (for example, see [4] for m∗ 1 and [5] for m∗ ∼ 1). The other approach is focused on the measurement of the fluid forces on an harmonically forced body (length L, cross-section side D) with amplitude A and frequency f in a uniform flow. Again, circular cylinders have been studied extensively [6–8]. In this case the lift force is usually represented by a force coefficient CY and a phase angle φ with the force leads the displacements, with the following dependence, CY , φ = Φ (A∗ , U ∗ , Re, L/D).
(2)
A. Barrero-Gil, P. Fernandez-Arroyo / European Journal of Mechanics B/Fluids 29 (2010) 364–368
The fluid forces acting on the body are usually measured by means of pressure taps located appropriately on the surface of the model or by means of piezoelectric gauges. The former method requires a large number of pressure taps for accuracy and the latter method requires that the inertial force FI (my¨ , being m and y¨ , respectively, the mass and acceleration of the body) must be subtracted from the total force registered by the piezoelectric gauge to obtain the fluid force. As the fluid forces FF are of order ρ U 2 D, m∗ must be of order unity for accuracy in the measurements [note that FF /FI ∼ U ∗ /(2π m∗ A∗ )]. That is the reason why these measurements are usually carried out in water facilities. However, another approach is possible when the fluid forces are determined from their effect on the motion of the mechanical system. In this case, the situation is more representative of the real fluid-structure interaction problem. However, to the author’s knowledge, this approximation has not been followed much by researchers of VIV. This is the strategy followed in this study. In the method proposed here a mathematical form with two undetermined parameters for the fluid excitation force is assumed. As will be shown later, they will be a function of the reduced velocity U ∗ and a least squares method will help us to identify these parameters from a designed experimental procedure. Fluid excitation means the fluid force component associated with the body velocity. For high mass ratio, the fluid force component in phase with cylinder acceleration (the added mass) has no practical importance in the VIV dynamics. This was clearly shown for the first time by Bearman [1]. Let us summarize his reasoning: a system such us that shown in Fig. 1 has a governing equation of motion given by 1
m(¨y + 4π ζ fN y˙ + 4π 2 fN2 y) =
ρ U 2 DCY (t ), (3) 2 where CY is the fluid force coefficient in the transverse direction. In the lock-in region (or resonance range), a good approximation of the fluid force and the cylinder displacement can be described as CY (t ) = CY sin(2π ft + φ),
(4)
y(t ) = A sin(2π ft ),
where f is the oscillation frequency and φ is the phase angle between the fluid force and the cylinder displacement. Substituting Eq. (4) in Eq. (3) and equating sine and cosine terms, the solution for the response amplitude and frequency is ∗
A =
f
∗
CY sin φ 16π 2 m∗ ζ
= 1−
U ∗2
f∗
CY cos φ U ∗2 8π 2 m∗ A∗
,
1/2
(5)
.
For a large mass ratio, m∗ 1, from Eq. (5) one can see that f ∗ −→ 1 (U ∗2 /8π 2 ∼ 1 and experiments show that CY 0 cos φ ∼ 1), and (U ∗ /f ∗ ) −→ St−1 (herein St is the stationary Strouhal number). Thus, one may deduce that response amplitude is a function of reduced velocity, mass-damping parameter (m∗ ζ ) and fluid force in phase with cylinder velocity (fluid excitation, CY sin φ ). Note that m∗ 1 is the situation in wind engineering and for most civilengineering structures, where air is the fluid that surrounds the body. In the following section, the experimental apparatus used to identify fluid excitation is described, as well as the experimental procedure. In Section 3, experimental results are presented. We show contours of fluid excitation as a function of the governing parameters (A∗ and U ∗ ), and a comparison with some previous experimental results is presented. Also, free vibrations experiments have been carried out to compare with the analytical prediction of the body response using the previously identified fluid excitation data. Finally, relevant results are summarized in Section 4.
365
Fig. 1. Flow-structure parameters in the VIV canonical problem.
2. Experimental details 2.1. Experimental set-up and procedure The work which is reported here has a main purpose: to measure fluid forces of an oscillating cylinder under cross flow by means of an inverse method. To this end, a spring-mass-dashpot system with an electromagnetic device to provide additional excitation has been designed. As can be seen in Fig. 2a, a circular cylinder of length L = 33 cm was supported elastically by means of two aluminium beams (40 cm long). The cylinder has a smooth surface finishing and is made out of plastic, and it was under the action of an airstream supplied by a small open-test section wind tunnel of rectangular area of 25 cm height, 40 cm width. The cylinder was situated 6 cm downstream of the test section entrance. In all cases, the width of the cylinder (D) was 3.3 cm, giving an aspect ratio (L/D) of 10, and the wind speed U was varied in the range 50–150 cm/s (so that, the Reynolds number, defined as Re = UD/ν , was varied between 1650 and 4950). The free-stream turbulence level on the centerline was below 2%. The transverse vibration, y(t ), was measured with a non-contact laser vibrometer (with a resolution of 60 µm), and the wind tunnel speed was measured with a hot wire anemometer (0.01 m/s of resolution). To provide accurate control in damping, a voice coil was used as actuator. The voice coil produces a force that is proportional to the current drive through the coil. If Va (t ) is the voltage across the coil, R it’s electrical resistance and x(t ) the displacement of the coil, then the force Fa supplied by the voice coil is given by (the coil inductance and capacitance can be neglected), Fa (t ) =
K R
Va (t ) −
K2 R
+ b x˙ ,
(6)
where K is a constant and b another one which takes into account the eddy current effects. If Va (t ) is proportional to the transverse velocity of vibration, y˙ (t ), and x(t ) is proportional to the transverse vibration amplitude of the square cylinder, then Fa (t ) = C y˙ (t ),
(7)
and therefore the voice coil introduces an external damping term in the system dynamics. As can be seen in the block diagram shown in Fig. 2b, after the elastic system block the output voltage of the laser vibrometer (proportional to y) is differentiated by means of a passive highpass filter with a cut-off frequency of 194 Hz (as the natural
366
A. Barrero-Gil, P. Fernandez-Arroyo / European Journal of Mechanics B/Fluids 29 (2010) 364–368
a
b Fig. 2. Experimental set-up (a) and schematic block diagram (b).
frequency of the system was 4.7 Hz the phase lag introduced is less than 2◦ ), so that V2 (t ) = RF CF (dVacc (t )/dt ) = k1 y˙ (t ) [RF and CF are, respectively, the resistance and capacity of the filter, k1 is a constant]. Then, V2 is amplified and sent to the power amplifier that drives the voice coil with a voltage proportional to the velocity of oscillation, Va (t ) = k1 k2 y˙ (t ) (and hence Fa = C y˙ ). The magnitude of the damping introduced, C , is varied with a potentiometer and its sign can be changed if the polarity of Va is reversed, providing accurate control of positive and negative damping. At the same time, the laser vibrometer signal is recorded by a PC at a rate of 500 Hz. To account for nonlinearities in the mechanical system (due principally to mechanical and fluid damping in still air) Eq. (3) is modified introducing a nonlinear term of the Van der Pol type (i.e. we assume that the damping force in the system can be represented as the sum of a linear damping term and a cubic damping term). Then, the dynamics of the system (see Fig. 2a) can be represented by the following equation,
tunnel is switched on at the desired velocity the voice coil is switch on and the parameters describing fluid excitation force, b1 (U ∗ ) and b2 (U ∗ ), are determined from the response by a least square fitting. For a particular velocity the test was repeated three times and the mean values were selected. The range of reduced velocity covered during the experiments was that within which VIV usually takes place, 3.5 < U ∗ < 10. Fluid excitation force was identified for eighteen values of reduced velocity. In all cases, the oscillation was registered over a 5 minute period (over 1400 cycles of oscillations). As shown before, the mechanical damping and fluid forces are smaller than the inertial and stiffness forces, and experiment shows that in the resonance range the response is essentially a sinusoidal function. Out of the resonance range, fluid forcing exhibits somewhat non-sinusoidal behaviour. However, as in previous studies [7,8], only the fluid excitation component (CY sin φ ) that occurs at the body oscillation frequency will be presented here. It is computed by means of a Fourier analysis of the identified fluid force.
m(¨y + 4π 2 fn2 y) + a1 y˙ + a2 y2 y˙ = FY (t ),
2.2. Post-processing and identification procedure
(8)
which represents a balance between the linear inertial and stiffness forces, the nonlinear mechanical damping force, and the fluid excitation force FY . Herein m is the equivalent mass per unit length of the system (mass of the cylinder plus the modal mass of the beam system), and fN is the natural frequency of the oscillations. First of all, a calibration of the installation was done. The stiffness k of the installation was measured by adding calibrated weights at both ends of the body and measuring its vertical displacement. Later, with the voice coil switch off, the body was excited with an initial displacement giving a frequency of oscillation of 4.7 Hz [and hence the equivalent mass per unit length of the system was deduced as m = k/(2π fn )2 ]. Finally, in still air (U ∗ = 0), the voice coil was switched on providing excitation (that is negative damping) for low amplitudes of vibration. The amplitude of oscillations grew until a limit cycle of oscillations around A∗ = 1 was reached. From the measured oscillations parameters a1 and a2 were computed by a least square fitting. The test was carried out three times and the mean values were selected. The repeatability of the procedure was fairly good, with a standard deviation of 0.018 (in the case of a1 ) and 0.028 (a2 ). To indirectly measure the fluid excitation force, FY (t ) is assumed to be described again by a Van der Pol term [9]. The rationale behind such a model is related to the experimental evidence that oscillations due to VIV have a self-limiting character, as were the electrical oscillations in the circuits studied by Van der Pol [10]. So that, FY (t ) = b1 y˙ + b2 y2 y˙ , where b1 (U ∗ ) and b2 (U ∗ ) are functions of the flow velocity (reduced velocity). After the wind
Putting Eq. (8) in a dimensionless form (η = y/D, τ = 2π ft) and taking into account that FY (t ) = b1 y˙ + b2 y2 y˙ , one obtains
η00 = −Ω η + k1 η0 + k2 η2 η0 ,
(9)
where Ω = (fn /f ) , k1 (U ) = (b1 (U ) − a1 )/(2π fmD), and k2 (U ∗ ) = (b2 (U ∗ ) − a1 )D/(2π fm). To find Ω (U ∗ ), k1 (U ∗ ), and k2 (U ∗ ) a least square fit has been used from the measured displacements (y, or in dimensionless form η) at the sampled points. At any instant Eq. (9) needs to be satisfied so that it can be expressed in a matrix form, 2
∗
∗
−→→ − →00 0− → η = [− η η2 η0 − η ][k1 k2 − Ω ]T = Aθ = b, (10) − → where θ = (k1 k2 Ω )T and b = η00 . Such a redundant system can be solved, in a least square sense, doing
θ = (AT A)−1 AT b.
(11)
Numerical differentiation was carried out using a centered five point method. For example, the velocity of oscillation, η0 , was computed as η0 (ti ) = [−η(ti+2 )+ 8η(ti−1 )− 8η(ti−1 )−η(ti−2 )]/(12∆); ∆ = ti − ti−1 being the sampling time. Next, a numeric filter with a cut-off frequency of fc /fn = 5 was applied to eliminate high frequency noise introduced by numerical differentiation. It should be noted that as it is necessary to compute the velocity and the acceleration of the cylinder carefully, the frequency of sampling must be as high as possible. In the experiments data were recorded at a rate of 500 Hz. Also, for better identification it is desirable that the
A. Barrero-Gil, P. Fernandez-Arroyo / European Journal of Mechanics B/Fluids 29 (2010) 364–368
367
Table 1 Comparison of the identified fluid excitation with the fluid excitation measured by Morse and Williamson [8] and Hover et al. [7] in forced oscillation tests. Comparison is only possible when the experimental conditions are similar. [8]: Re = 4000, L/D = 10. [7]: Re = 3800, L/D = 19.2. A∗
Morse and Williamson [8]
Present data
Re
U∗
CY sin φ
Re
U∗
CY sin φ
0.55 0.65 0.75 0.9
4000 4000 4000 4000
7.87 7.87 7.87 7.87
0.1 −0.2 −0.4 −0.6
4000 4000 4000 4000
7.87 7.87 7.87 7.87
−0.25 −0.45 −0.8
A∗
Hover et al. [7]
0.5 0.65 0.9 1
0.03
Present data
Re
U∗
CY sin φ
Re
U∗
CY sin φ
3800 3800 3800 3800
7.43 7.43 7.43 7.43
0
3800 3800 3800 3800
7.43 7.43 7.43 7.43
−0.26 −0.65 −0.9
−0.2 −0.4 −0.6
0.1
Table 2 Estimated values for different noise levels (or relative error). True values
Estimated values
= 0.01 k1 = 0.01 k2 = −0.045
= 0.05
= 0.1
0.0098
0.0097
0.0090
−0.0438
−0.0413
−0.0386
experimental setup presents low values of a1 (<0 and controlled by the actuator) and a2 (i.e. that the mechanical system presents a small nonlinearity). 2.3. Experimental uncertainties Let us now to discuss the main sources of uncertainties in our measurements. The laser displacement used (M7 from MEL Mikroelektronik) had a resolution of ±0.06 mm, and therefore if the oscillation amplitude of the cylinder was higher than A∗ = 0.05 the measured displacement is known to within an error of less than 3.6%. The resolution of the hot wire was ±0.01 m/s and during experiments the mean flow velocity measured was known within an error of the order of 1%. On the other hand, the diameter of the cylinder was measured with a vernier caliper within an error of 0.25% and the weights added to calculate the stiffness of the mechanical system were measured with a balance to a resolution of ±0.1 g (relative error always less than 2%). Then, taking into account the error propagation, the normalized velocity A∗ was known within a nominal error of the order of 4% and the reduced velocity within a nominal error of 3.5%. The mode in which uncertainties in the measurement of the displacement are propagated in the velocity and acceleration computations (and hence in the identified parameters) is not easy to estimate. To test the influence of uncertainties in the estimated parameters a sequence of white noise was added to simulated data. These data were obtained numerically solving Eq. (9) with a fourth order Runge–Kutta algorithm. Table 2 shows some typical values for various noise levels. It seems that the estimation is sensitive to the influence of noise and coefficients are poorly estimated for high noise. 3. Results 3.1. Fluid excitation The identified fluid excitation coefficient is shown in Fig. 3 in the parameter space (U ∗ , A∗ ). With comparison purposes values from the experiments carried out by Morse and Williamson and Hover et al. are also presented in Table 1. The contour of zero fluid excitation (CY sin φ = 0) is particularly interesting. It divides the region in the (U ∗ A∗ ) plane where energy transfer is from the flow to the body (CY sin φ > 0; excitation)
Fig. 3. Contour plot of the identified fluid excitation, CY sin φ . Contour interval is 0.2. Note that x axis stands for the normalized velocity as well as the Reynolds number (upper one). There is a clear correspondence, as Re = U ∗ (fN D2 /ν), where β = fN D2 /ν is the Stokes number (=511 in this case). Also, three distinct zones (corresponding to initial, upper,and lower branch) are identified at the top of the figure.
from that where the energy transfer is from the body to the flow (CY sin φ < 0). With respect to its shape, we can distinguish four different zones or, following the nomenclature introduced by Khalak and Williamson [5], branches. The initial branch (3.5 < U ∗ < 5), where fluid excitation increases with reduced velocity, is followed by the upper branch (5 < U ∗ < 6.5) where some hysteretic behavior can be seen, as fluid excitation is not univocally determined by normalized amplitude (for example, for U ∗ = 5.5 three possible values of CY sin φ can be distinguished). For (6.5 < U ∗ < 9) fluid excitation is fairly constant. In the literature this is known as the lower branch. Finally, for U ∗ < 9 fluid excitation drops rapidly. On the other hand, the zero fluid excitation boundary represents the maximum amplitude of oscillation achievable during VIV (when the mass damping parameter is very low). In this case at a Reynolds number of 2933 (U ∗ = 5.74) and m∗ ζ −→ 0, the maximum amplitude of oscillation would be of 0.81. The value proposed by Govardham and Williamson [11] for the above mentioned Reynolds number is 0.85, close to the one proposed here. 3.2. Free vibration experiments To test the identified fluid excitation force, the same experimental arrangement without the action of the voice coil was used to look for response amplitude versus reduced velocity. The experimental protocol was always the same: fixing the airstream velocity, the cylinder was exposed over a sufficiently long time until a steady state of oscillation was reached; then, the oscillation was registered during 60 s. Next, the flow velocity was increased by a small amount with the cylinder still oscillating and the operation was repeated. Experimental results are compared to analytical predictions with the identified values of fluid excitation, as shown in Fig. 4. From the figure it can be seen that the maximum amplitude of oscillations is well captured, as well as the reduced velocity which occurs. Also, the range of reduced velocities where oscillations are of significance is well captured. It should be noted that analytical values can be obtained solving iteratively Eq. (5): for a fixed reduced velocity, the normalized
368
A. Barrero-Gil, P. Fernandez-Arroyo / European Journal of Mechanics B/Fluids 29 (2010) 364–368
knowledge of the fluid excitation component of the force could be enough to describe VIV until values of the mass ratio as low as 10. 4. Conclusions
Fig. 4. Comparison of measured free vibration (black circles) and predicted with the identified fluid excitation force (dashed line). m∗ = 260, ζ = 0.0024.
The determination of fluid forces for an oscillating circular cylinder in cross flow is a problem of engineering significance, as its VIV response can be estimated with the aid of measured fluid forces. In this paper, an experimental procedure to identify fluid excitation on a spring-mounted circular cylinder in cross flow has been presented. While we can control linear damping on the experimental arrangement, an accurate identification of fluid excitation is possible. Contrary to the traditional employed case of forced oscillations, fluid excitation is identified from a fluidstructure interaction experimental set-up, creating a more realistic approach to the problem. We have assumed a nonlinear form for fluid excitation and identified it for a circular cylinder of aspect ratio of 10 in the reduced velocity range of 3.5 < U ∗ < 10 (1790 < Re < 5110) and normalized amplitude 0 < A∗ < 1. Results show good correspondence with those obtained from previous forced oscillations experiments. Also, the identified fluid excitation has predicted well the free vibrations of the same elastically supported circular cylinder, and comparison with free vibrations for other experimental studies is good. Finally, it must be noted that the technique here described can be used to identify fluid excitation (and hence VIV response) on other geometries prone to VIV with high interest from the civil engineering point of view like, among others, square or rectangular prisms and H-section prims. In these cases, as the separation of the flow generally occurs at fixed points, the dependence upon the Reynolds number is expected to be weaker that for the case of a circular cylinder. Acknowledgements The authors wish to thank the referees for their valuable comments. References
Fig. 5. Comparison of free vibration data from [12] (open circles) with the predicted values using the identified fluid excitation force (solid line).
amplitude is increased from zero (and thus the fluid excitation coefficient evaluated) until a value where Eq. (5) is satisfied within a prescribed tolerance. 3.3. Prediction for moderate m∗ It was shown in the introduction that fluid excitation (CY sin φ ) very approximately governs the dynamics of VIV for high mass ratio values. However, Fig. 5 shows experimental results of the A∗ vs. U ∗ behaviour from [12] for m∗ = 7.1, ζ = 0.0014, β = fn D2 /ν = 520, together with analytical predictions with our identified values of fluid excitation (where β = 511). As can be seen, the agreement is not so bad. The peak amplitude predicted is close to the one measured by Klamo and the range of resonance is similar in both cases. We think that this is a somewhat expected result. Maull and Kaye [13], in their experiments where m∗ ' 12 found only a very little shift of resonances of the response of a flexible cylinder in waves (f ∗ was very close to one). On the other hand, the range of resonance predicted by the fitting formula introduced by Govardham and Williamson [14] is quite constant for m∗ > 10. Thus, one might expect that fluid excitation governs approximately the A∗ vs. U ∗ behavior until m∗ ' 10. Then, the
[1] P. Bearman, Vortex shedding from oscillating bluff bodies, Ann. Rev. Fluid Mech. 16 (195) (1984) 222. [2] T. Sarpkaya, A critical review of the intrinsic nature of vortex-induced vibrations, J. Fluids Struct. 19 (389) (2004) 447. [3] C.H. Williamson, R. Govardhan, Vortex-induced vibrations, Ann. Rev. Fluid Mech. 36 (413) (2004) 455. [4] C.C. Feng, The measurement of vortex induced effects in flow past a stationary and oscillating circular and D-section cylinder, M.A.Sc. Thesis, University of British columbia, Canada, 1968. [5] A. Khalak, C.H. Williamson, Dynamics of a hydroelastic cylinder with very low mass and damping, J. Fluids Struct. 10 (1996) 455–472. [6] T. Staubli, Investigation of oscillating forces on a vibrationg cylinder in cross flow, Doctoral dissertation, ETH Zurich, 1983. [7] F.S. Hover, A.H. Techet, M.S. Triantafyllou, Forces on oscillating uniform and tapered cylinders in cross flow, J. Fluid Mech. 363 (1998) 97–114. [8] T.L. Morse, C.H. Williamson, Fluid forcing, wake modes, and transitions for a cylinder undergoing controlled oscillations, J. Fluids Struct. (2009) doi:10.1016/j.jfluidstructs.2008.12.003. [9] E. Simiu, H. Scanlan, Wind Effects on Structures, Wiley, New York, 1996. [10] R.D. Gabbai, H. Benaroya, An overview of vortex-induced vibration of circular cylinders, J. Sound Vibration 282 (2005) 575–616. [11] R. Govardham, C.H. Williamson, Defining the ‘modified Griffin plot’ in vortexinduced vibration: revealing the effect of Reynolds number using controlled damping, J. Fluid Mech. 561 (2006) 147–180. [12] J.T. Klamo, Effects of damping and Reynolds number on vortex-induced vibrations, Ph.D. Thesis, Caltech, 2007. [13] D.J. Maull, D. Kaye, Oscillations of a flexible cylinder in waves, in: Proc. 5th Conf. on Behaviour of Offshore Structures, BOSS, Trondheim, vol. 2, 1988, pp. 535–547. [14] R. Govardham, C.H. Williamson, Modes of vortex formation and frequency response from a freely-vibrating cylinder, J. Fluid Mech. 420 (2000) 85–130.