Response characteristics of a vortex-excited circular cylinder in laminar flow

Transcription

1 Journal of Mechanical Science and Technology 25 (1) (2011) 125~133 DOI /s sponse characteristics of a vortex-excited circular cylinder in laminar flow M. H. Bahmani and M. H. Akbari * School of Mechanical Engineering, Shiraz University, Shiraz , Iran (Manuscript ceived February 20, 2010; vised July 6, 2010; Accepted October 4, 2010) Abstract This paper presents numerical simulation results for vortex-induced vibration of a circular cylinder in laminar flow. A vortex method is implemented to solve the two-dimensional Navier-Stokes equations in terms of vorticity. In order to validate the numerical code, the flow past a fixed cylinder is first investigated for which enough experimental and numerical results are available. Basic characteristics of the dynamic response and vortex shedding for an elastically mounted circular cylinder are then investigated for 70 < < 170. The lock-in phenomenon is captured at certain reduced velocities where the lift coefficient takes a considerable value associated with a high amplitude response. The wake structure exhibits the 2S or C (2S) modes of vortex shedding in this range of ynolds numbers, as opposed to the 2P mode which is observed in the turbulent flow regime. The numerical results are in acceptable agreement with available experimental and numerical data. Keywords: Circular cylinder; sponse characteristics; Vortex-induced vibration; Vortex method Introduction Uniform flow past a circular cylinder is an important and fundamental subject in fluid mechanics. It is well known that flow behind a stationary cylinder becomes unstable beyond a specific ynolds number, where vortex shedding starts to appear. Vortex shedding results in fluctuating lift and drag forces, which will cause the cylinder to vibrate if it is elastically mounted. This phenomenon is known as vortex induced vibration (VIV). VIV of a circular cylinder is of great importance in many engineering fields, such as heat exchangers, offshore structures, towers, and bridges. The practical significance of uniform flow past a circular cylinder either when it is stationary or vibrating has lead to it being largely studied. Some researchers classified different regimes of flow past a stationary cylinder (e.g., Chen [1]; Lienarhard [2]). Many researchers studied the fluid forces on the cylinder and the frequency of vortex shedding for different regimes (e.g., Williamson [3]). Feng [4] showed that an elastically mounted cylinder may exhibit a range of responses depending on a number of parameters, including reduced velocity U * (U * =U/ f n D, where U, D and f n are flow velocity, cylinder diameter, and natural frequency of the system, respectively), ynolds number, This paper was recommended for publication in revised form by Associate Editor Kyung-Soo Yang * Corresponding author. Tel.: , Fax.: address: h-akbari@shirazu.ac.ir KSME & Springer 2011 mass ratio m * (ratio of cylinder mass to displaced fluid mass, m * = (4m)/ (πρd 2 L)), and damping ratio ζ, (ratio of damping coefficient to critical damping coefficient, ζ=c/c critical ). Later, many researchers studied the variations of system responses to the change of these parameters (e.g., Khalak and Williamson [5]; Klamo et al. [6]). Some of them suggest that the system response is characterized by combined mass-damping, m * ζ [7, 8]. On the other hand, some researchers state that the system response is influenced by the independent effects of mass and damping [9]. One of the most important characteristics in the case of an oscillating cylinder is a lock-in phenomenon in which the vortex shedding and oscillation frequencies synchronize. This synchronization can lead to an amplification of the cylinder s vibrational response. Some researchers classified different regimes of synchronous response and attempted to explain the differences between these modes. Khalak and Williamson [10] conducted a series of experiments and studied different regimes of synchronization and response of the system. Later, Khalak and Williamson [11] and Govardhan and Williamson [12] continued their investigation. In addition to Karman street vortex shedding from a stationary cylinder, one may face more complex vortex patterns in the wake of a vibrating cylinder. Williamson and Roshko [13], for example, classified the case with two single vortices shed per cycle as the 2S regime, and the case with two vortex pairs shed per cycle as the 2P mode, while in their classification P+S type indicated a pattern in which one pair and a single vortex are shed in each cycle.

2 126 M. H. Bahmani and M. H. Akbari / Journal of Mechanical Science and Technology 25 (1) (2011) 125~133 Anagnostopoulos and Bearman [14] studied experimentally free vibration of a circular cylinder in laminar flow. They found that the oscillation amplitude is stabilized after many oscillation cycles within the lock-in region. They also found that the cylinder oscillates with a frequency slightly higher than its natural frequency in air in the greatest part of the lockin region. In most of the VIV simulations, the cylinder is free only in the lateral direction. Some researchers studied the in-line VIV of a cylinder or when the cylinder has two degrees of freedom (e.g., [9,15,16]). Jauvtis and Williamson [17] studied the behaviour of an elastically-mounted cylinder with both streamwise and transverse motions. The range of values of m * in their study was between 5 and 25. It was found that the freedom of body to oscillate in both directions has very little effect on the transverse response of the system and the vortex wake dynamics. Pastò [18] investigated experimentally VIV of a circular cylinder in laminar and turbulent flows. He studied the effects of cylinder roughness and mass-damping parameter, m * ζ, on the system response and proposed that the mass-damping parameter is not the sole parameter influencing the response; the ynolds number also plays an important role. With the increase in the power of computers, many researchers studied the VIV of circular cylinder numerically. Newman and Karniadakis [19], using direct numerical simulation, investigated the flow over a flexible cable. In particular, they compared forced cable vibration with VIV at ynolds numbers of 100, 200 and 300. Singh and Mittal [20] simulated numerically VIV of a circular cylinder at low ynolds numbers () for a system of low non-dimensional mass (m * =10) and with a structural damping coefficient of zero. They carried out two sets of computations to investigate the effect of and reduced natural frequency, F n (F n =1/U * ), independently. They found that the effect of is very significant. Studying previous investigations of the VIV of circular cylinders, one can notice some important features: first, it is observed that most of these studies are at ynolds numbers where the flow is inherently turbulent and three dimensional; second, it appears that there are many discrepancies between predictions from computational results and even between the empirical methods leading to experimental data (Chaplin et al. [21]). These differences are thought to be due to the intrinsic nature of the dynamic system, its capacity to vibrate at high mode numbers and the complex added mass and hydrodynamic damping distributions while the body is in motion. In the present study, the VIV of a circular cylinder with low damping in laminar flow is simulated using a two-dimensional CFD code; the use of a two-dimensional Navier-Stokes solution method is reasonable because the flow is basically twodimensional in this range of the ynolds number. The vorticity filed is also investigated particularly in the synchronization zone. One rare experimental investigation we found in the literature for free vibration of a cylinder in laminar flow was conducted by Anagnostopoulos and Bearman [14]. Moreover, there are comparatively few low ynolds number VIV simulations among which Anagnostopoulos [22] could compare his numerical results with their previous experimental data. It was therefore decided to first numerically simulate their experiments and investigate different aspects of the flow field. The unsteady nature of the flow field is the main cause of VIV; thus, in predicting VIV of an object, an accurate prediction of the flow field is essential. Anagnostopoulos and some other researchers used a finite element scheme [15, 22, 23]; however, in the present simulation, a vortex method is implemented which is also a Navier-Stokes solver. In this method, the vorticity filed is discretized into point vortices which are convected by the local velocity and are diffused due to viscous effects of the flow field. Vortex methods are believed to be very suitable for separated flow fields such as those around bluff bodies [24, 25]. This method was chosen because of its several advantages. First, it is possible to extend the physical domain to hundreds of diameters from the cylinder without much increase in the computational time or memory requirements. This domain is several times larger than those in previous studies such as those by Anagnostopoulos [22]. Secondly, this method is not iterative and hence is not involved with convergence problems. Finally, because of a comparatively low computational time, we are able to extend the time domain to obtain the steady response of the system. Our study is therefore organized in two parts. In the first part, the flow around a stationary cylinder is investigated for which many experimental results are available. Different characteristics of the flow field as well as fluid forces on the cylinder are calculated for this case. This part will verify the accuracy of our CFD code. In the second part, the VIV of a cylinder in laminar flow is investigated, and basic characteristics of the dynamic response and vortex shedding from the cylinder are studied. Our results are compared with previous experimental and numerical studies, and a good agreement is obtained. 2. Numerical approach The governing equations for the incompressible flow of a Newtonian fluid are the Navier-Stokes and continuity equations. The Navier-Stokes equations can be expressed in terms of vorticity and stream-function as follows: + t r r ( u. ) ω = ν ω ω 2 where ω and u r are the vorticity and velocity vector, respectively, and ν is the kinematic viscosity of the fluid. Given the vorticity field, the Poisson equation 2 ψ = ω can be solved for the stream functionψ, the spatial derivative of which gives the velocity components. To solve the vorticity equation, Eq. (1), the flow field may be represented by point vortices in a vortex method. The point vortices are originally created to (1)

3 M. H. Bahmani and M. H. Akbari / Journal of Mechanical Science and Technology 25 (1) (2011) 125~ satisfy the no-slip condition. The governing equations of the flow may then be solved by considering the creation, interaction and motion of these point vortices. An operator splitting method is used to solve Eq. (1), where this equation is split into convection and diffusion parts as follows: ω r r = ( u. ) ω t (2) ω = ν 2 ω. t (3) The time domain is discretized into small time-steps, during which Eqs. (2) and (3) are solved sequentially. The numerical algorithm implemented to solve these equations is based on a vortex method described in detail by Akbari and Price [26, 27] and Bahmani and Akbari [28], and are not presented here for the sake of brevity. In the present vortex method, a computational grid is used for the following purposes: solving the Poisson equation for the stream-function given the vorticity field, calculating the velocity field given the streamfunction, and (c) solving the diffusion equation for the vorticity field. New vortices are created during each time step at the location of the first row of the grid nodes, on the surface of the body. The computational grid is fixed to the body, and hence, it moves with the cylinder as it vibrates. Once the governing equations are written in terms of vorticity, the pressure term is eliminated from the equations. In order to calculate the surface pressure, the Navier-Stokes equation in the radial direction is used. The pressure gradient can be obtained from finite difference approximations of the velocity variation, with pressure at the outer limit of the mesh taken as constant everywhere. The transverse response of the cylinder, y (t), to the fluid force loading per unit span, F y (t), is modeled using a mass, spring and damper model. The cylinder's equation of transverse motion is thus given by: m & y + c y& + k y = F (t) (4) y where m is the cylinder mass, c the damping coefficient, k the structural stiffness, y the transverse cylinder displacement, and F y is the transverse fluid force on the cylinder. Having calculated the transverse force, a fourth-order Runge-Kutta scheme is used to solve Eq. (4). 3. sults In this investigation a numerical algorithm was developed to simulate vortex-induced vibration of an elastically mounted circular cylinder. Undoubtedly, success of such simulation depends on an accurate modelling of the flow field. Our study is therefore organized in two parts. In the first part, the flow around a stationary circular cylinder is investigated. The purpose of this part is to make certain that the simulation code can model such unsteady flow field with acceptable accuracy. In the second part, the VIV of a circular cylinder with low damping is simulated; this system was experimentally analyzed previously by Anagnostopolous and Bearman [14]. The present results are also compared with some previous numerical data [22, 29]. The simulation results of this part also show good agreement with the experimental and numerical data. 3.1 Stationary cylinder In this part, different characteristics of the flow field and the fluid forces on a stationary cylinder are investigated. This will allow us to verify the capability of the present code in predicting such unsteady flow field at low ynolds numbers. The wake behind a circular cylinder exhibits two types of behaviors in this range of the ynolds number. For very low ynolds number flows, the wake behind the cylinder is steady; but once the ynolds number is increased beyond a specific value the wake becomes unstable and vortex shedding starts to occur. The present numerical results for the length of the recirculation zone versus are shown in Fig. 1, where a comparison is made with the experimental data of Taneda [30] and Tritton [31]. The simulation results fall within 5% of the corresponding experimental data. Also, the variation of the drag coefficient with is shown in Fig. 1. It is observed that by increasing the ynolds number, the length of the recirculation zone increases and the drag coefficient decreases. Good agreement is found between the present numerical results and previous experimental data. Once the ynolds number is increased beyond 50, the wake behind the cylinder becomes unsteady and vortex shedding starts to occur. Fig. 2 shows the vorticity contours behind the cylinder at = 100. Also shown in Fig. 2 is the time history of the lift and drag coefficients for this case. It is seen that after some transient response associated with the impulsive starting of the flow, the flow reaches an approximately periodic state where the lift and drag forces settle to regular periodic variations. The frequency of the vortex shedding is usually stated in nondimentinal form by the Strouhal number which is defined as St = f v D/U, where f v is the vortex shedding frequency, and D and U are the cylinder diameter and the free stream velocity, respectively. To find the frequency of the vortex shedding a power spectral analysis is used, the result of which is shown in Fig. 2(c). Other cases in the range 60 < < 200 were studied and the results are summarized and compared with the results of previous studies in Fig. 3. Part of the figure presents the Strouhal-ynolds curve, where a comparison is made with the universal Strouhal-ynolds relation given by Williamson [3]. It is seen that the predicted variation of St with agrees quite well with the experimental data. Figs. 3 and 3 (c) show that the lift coefficient amplitude increases with the

4 128 M. H. Bahmani and M. H. Akbari / Journal of Mechanical Science and Technology 25 (1) (2011) 125~ experimental results Present numerical results L r / D C d experimental results 3 present numerical results Fig. 1. Variation with of length of recirculation zone and drag coefficient for the steady-wake regime ( < 50) and comparison with experimental data: ( ) present results; ( ) Taneda [30]; ( ) Tritton [31]. ynolds number, while the mean drag coefficient decreases. It is clear that the accuracy of our results is quite acceptable for these characteristics of flow field as well. The data presented in this section confirm the capability of the present numerical approach in predicting the unsteady flow behind a circular cylinder in the laminar flow regime with adequate precision. 3.2 Free vibrating cylinder (c) Fig. 2. Simulation results for the cross flow around a stationary circular cylinder at = 100 vorticity contours; time histories of C l and C d ; (c) power spectrum diagram of C l. Investigated in this section is the system response associated with vortex-induced vibration of a circular cylinder with low damping in laminar flow. Most of previous studies of VIV of a cylinder have been at ynolds numbers where the flow is inherently turbulent and three dimensional. One rare experimental investigation found in the literature for free vibration of a cylinder in laminar flow is by Anagnostopoulos and Bearman [14]. It was therefore decided to first simulate numerically their experiments, and investigate different aspects of the flow field. The physical parameters were set according to the experiments as m * =149.1, f n =7.016, ζ= The reduced velocity range is 4 to 9.5. The changes in reduced velocity are achieved by altering the flow velocity; the ynolds number will also change according to =17.9U *. Hence, the ynolds number changes from 70 to 170 in these simulations. For each reduced velocity between 4 and 9.5, the time history of transverse displacement and lift force as well as the vortex shedding and oscillation frequencies are obtained. By comparing the vortex shedding and oscillation frequencies, the reduced velocity band over which the lock-in phenomenon occurs is found. Fig. 4 shows the time history of the system response and lift coefficient of the cylinder for different cases with various reduced velocities (t * is the non-dimensional time, t * = t U / D). These four cases are chosen to show the onset of lock-in and how the system response and the fluid force change when synchronization occurs. The displacement and lift coefficient histograms for the case with U * =5.03 (=90) are shown in part of the figure. As seen, the oscillation amplitude of the cylinder is very small in this case. It is also observed that the time variation of the lift coefficient is similar to that for a stationary cylinder with its maximum value slightly higher. Since the vortex shedding and oscillation frequencies are different, lock-in did not occur in this case. A beating effect is present in the displacement and lift coefficient histograms for U * =5.03 (=90), which is in agreement with the experimental results by Anagnostopolous and Bearman [14].

5 M. H. Bahmani and M. H. Akbari / Journal of Mechanical Science and Technology 25 (1) (2011) 125~ St C ' l C d (c) Fig. 3. Variation of with of Strouhal number; amplitude of lift coefficient; (c) mean drag coefficient for a stationary circular cylinder in laminar flow and comparison with previous experimental and numerical data: ( ) present results; ( ) Williamson [3]; ( ) Placzek et al. [32]; ( ) Henderson [33]. Increasing the reduced velocity to U * =5.59 (=100), it is observed in Fig. 4 that the amplitude of oscillation has slightly increased. The amplitude of lift coefficient is slightly higher than that of a fixed cylinder. Lock-in does not occur in this case either, and a beating effect is present in this case as well. Increasing slightly the reduced velocity to U * =5.87, (=105), we face a sudden change in the system response as shown in Fig. 4(c). The oscillation amplitude increases gradually before it reaches a steady value after many oscillation cycles. It is also seen in this case that the vortex shedding frequency diverges from that for a stationary cylinder and synchronizes with the cylinder oscillation frequency, i.e. lock-in has occurred. ferring to the time history of the lift coefficient in this case, it is observed that the amplitude of the lift coefficient has increased considerably. Fig. 4(d) shows the system response for a case with U * =6.15 (=110). Comparing the vortex shedding frequency and cylinder oscillation frequency, we see that lock-in occurs in this case, and consequently the amplitude of oscillation increases for many oscillation cycles until it reaches a steady value. The amplitude of the lift coefficient also increases but eventually reaches a steady state. The results presented in Fig. 4 indicate that the response of the cylinder, in particular its amplitude of oscillation and the lift coefficient, varies significantly when the reduced velocity is changed. Power spectrum analyses of the cylinder displacement for these cases show that in all cases the cylinder oscillates with a frequency near the natural frequency of the cylinder. It must be noted that the time scales in parts to (d) of Fig. (4) are not the same; hence the oscillation frequency in these cases may not be seen as close as they actually are. For the cases in the lock-in region, the time domain is extended further to obtain the steady system response. The displacement and frequency response of the cylinder, as well as its lift coefficient, were obtained for a range of ynolds numbers in the range of 70 and 170, and the results are summarized in Figs. 5 to 7. Fig. 5 shows the variation of the maximum non-dimensional amplitude of the cylinder with reduced velocity (ynolds number), where a comparison is made between the present numerical data and the results of previous studies. It is observed that the general trend of the lock-in phenomenon is captured well in our numerical simulations. It is seen that the reduced velocity range in which lock-in occurs and the high amplitude response of the system are in good agreement with experimental results (see Fig. 6). Unlike Anagnostopolous and Bearman (1992) that reported the maximum amplitude near the lower limit of the synchronization zone, we found the peak amplitude of oscillation somewhere in the first half the lock-in region. This, however, is in agreement with the results of some previous studies (e.g., Pastò [18], Leontini et al. [34] and Shiels et al. [29]). Therefore it was decided to compare the results of the present investigation with the previous numerical data. Additional simulations were performed for the cases investigated by Shiels et al. [29]. The system parameters are set accordingly as: =100, m* =5 and ζ=0. The amplitude of the system response was obtained as a function of nondimensional velocity (U R = U/2πf n D). Fig. 6 shows a comparison of the results, where a good agreement is observed between the present data and those given previously by Shiels et al. [29]. It is seen that in both sets of the numerical data, the maximum oscillation amplitude has been predicted somewhere in the first half of the lock-in region. This agreement gives us confidence in the accuracy of our results. Fig. 7 shows the variation of the cylinder oscillation frequency, f c and the vortex shedding frequency, f v with the ynolds number as obtained in the present study. A comparison is also made with the corresponding experimental results of Anagnostopolous and Bearman [14]. The dotted line shows

6 130 M. H. Bahmani and M. H. Akbari / Journal of Mechanical Science and Technology 25 (1) (2011) 125~133 (c) Fig. 4. Displacement and lift coefficient histogram at different reduced velocities for a free-vibrating cylinder U * =5.03 (=90); U * =5.59 (=100); (c) U * =5.87 (=105); (d) U * =6.15 (=110). (d) the frequency of vortex shedding for a stationary cylinder. It is observed that in a range of reduced velocity, the vortex shedding frequency diverges from that for a stationary cylinder and synchronizes with the cylinder oscillation frequency (lock-in). It is noted that outside the lock-in zone, the vortex shedding frequency is near that of a stationary cylinder. Similar to the cylinder amplitude (Fig. 5), the reduced velocity at the onset of lock-in and the reduced velocity band

7 M. H. Bahmani and M. H. Akbari / Journal of Mechanical Science and Technology 25 (1) (2011) 125~ Fig. 5. Variation of the maximum non-dimensional cylinder oscillation amplitude with U * () for a free-vibrating cylinder :, present numerical results; Experimental results (Anagnostopoulos and Bearman, [14]) A/D Fig. 6. Variation of the maximum non-dimensional cylinder oscillation amplitude with U R for a free-vibrating cylinder:, present numerical results; previous numerical results (Shiels et al. [29]). over which lock-in occurs are in agreement with the experimental results of Anagnostopoulos and Bearman [14]. The variation of the root mean square of the lift coefficient (C l r.m.s ) with reduced velocity (ynolds number) is summarized in Fig. 7. The lift coefficient shows a considerable increase in the lock-in zone which is associated with the high amplitude response. The value of the lift coefficient outside the lock-in region is slightly higher than that of a stationary cylinder. The vorticity field for each case was obtained and studied. The vorticity contours in the wake of a free vibrating cylinder at all reduced velocities considered here exhibited vortex shedding of 2S type (two opposite vortices are shed per cycle). However, there are some differences between the cases inside and outside of the lock-in region. For the reduced velocities where lock-in did not occur, the amplitude of oscillation is small and the vorticity contours are very similar to those for a fixed cylinder at the same ynolds number. Once lock-in occurs and the cylinder experiences a high amplitude oscillation, the vorticity field structure is a little different from that for a fixed cylinder at the same ynolds number. It is observed that when lock-in occurs vortex spacing is smaller and U R Fig. 7. Variation of the frequency response and lift coefficient with reduced velocity and ynolds number for a free-vibrating cylinder. (The frequency response is compared with experimental results of Anagnostopoulos and Bearman, [14]). the vortices in the far wake coalesce, shifting towards the C (2S) mode of vortex shedding. Instantaneous vorticity contours for a sample case in the lock-in region with U * =6.13 (=110) are shown in Fig. 8 at several instances after the start of the free vibration of the cylinder. It is seen that at the start of the free vibration, vortex shedding is very similar to that of a stationary cylinder because the amplitude of oscillation is very small (Fig. 8). As time passes and the amplitude of oscillation increases gradually, vortices shed from the cylinder surface move closer to the cylinder and become stronger (Figs. 8 and (c)). With further passage of time, the vortex shedding and cylinder oscillation frequencies synchronize and the cylinder experiences a high amplitude free vibration. In this case, the vortex shedding mode diverges from the standard 2S type, and vortices start to coalesce in the far wake (Figs. 8(d) and (e)). This is similar to what Williamson and Roshko [13] observed in their experiments and called the C (2S) mode. The C (2S) mode is very similar to the 2S mode except that vortices coalesce in the far wake. Fig. 8 (f) shows the vorticity field for the time when the vibration amplitude has reached a steady state and the vortex coalescence occurs in a closer distance to the cylinder.

8 132 M. H. Bahmani and M. H. Akbari / Journal of Mechanical Science and Technology 25 (1) (2011) 125~133 It is observed that the lift coefficient has a considerable value in the lock-in region which is associated with a high amplitude response. The wake structure was studied as well, where it was observed that the 2S or C (2S) modes of vortex shedding were present in this range of the ynolds number (unlike in the turbulent flow regime where the 2P mode is observed). This is in good agreement with the experimental results of Williamson and Roshko [13] and the numerical results of Singh and Mittal [20] and Prasanth and Mittal [29]. (c) (d) (e) (f) Fig. 8. Instantaneous vorticity contours for a free-vibrating circular cylinder at U * =6.13 ( = 110): t * = 10; t * = 150; (c) t * = 300; (d) t * = 500. (e) t * = 800; (f) t * = Summary and conclusion In this paper, the vortex induced vibration of a circular cylinder at low ynolds numbers was numerically investigated. In order to validate the accuracy of the developed numerical algorithms, the flow around a fixed cylinder was first studied and different characteristics of the flow field were verified against previous results. The present results showed that our numerical method is capable of simulating the flow field with acceptable accuracy. The VIV of an elastically mounted circular cylinder in laminar flow was then investigated. This problem was previously analyzed experimentally by Anagnostopoulos and Bearman [14]. The present simulations accurately capture the lock-in phenomenon and its characteristics, including the reduced velocity at the onset of lock-in and the reduced velocity range over which lock-in occurs. ferences [1] S. S. Chen, Flow-Induced Vibration of Circular Cylindrical Structures, Springer-Verlag, Berlin (1987). [2] J. Lienhard, Synopsis of Lift, Drag and Vortex Frequency Data for Rigid Circular Cylinder, Washington State University, College of Engineering, search Division Bulletin 300, [3] C. H. K. Williamson, Defining a universal and continuous Strouhal-ynolds number relationship for laminar vortex shedding of circular cylinder, Phys. Fluids, 31 (1988) [4] C. C. Feng, The Measurement of Vortex-Induced Effects in Flow Past Stationary and Oscillating Circular and d-section Cylinders, M.Sc. Thesis, University of British Columbia, Canada (1968). [5] A. Khalak and C. K. H. Williamson, Investigation of relative effects of mass and damping in vortex-induced vibration of a circular cylinder, J. Wind Eng. Indust. Aerodynamics, (1997) [6] J. T. Klamo, A. Leonard and A. Roshko, The effects of damping on the amplitude and frequency response of a freely vibrating cylinder in cross-flow, J. Fluids Structures, 22 (2006) [7] O. M. Griffin, Vortex-excited cross flow vibrations of a single cylindrical tube, J. Pressure Vessel Technol., 102 (1980) [8] E. Naudascher and D. Rockwell, Flow-Induced Vibration: An Engineering Guide, Balkema, Rotterdam, Netherlands (1993). [9] T. Sarpkaya, Hydrodynamic damping, flow-induced oscillation, and biharmonic response, ASME J. Offshore Mech. Arctic Eng. 117 (1995) [10] A. Khalak and C. K. H. Williamson, Dynamics of a hydrostatics cylinder with very low mass and damping, J. Fluids Structures, 10 (1996) [11] A. Khalak and C. K. H. Williamson, Motion, forces and mode transitions in vortex-induced at low mass-damping, J. Fluids Structures, 13 (1999) [12] R. Govardhan and C. H. K. Williamson, sonance forever: existence of a critical mass and an infinite regime of resonance in vortex-induced vibration, J. Fluids Structures, 473 (2002) [13] C. H. K, Williamson and A. Roshko, Vortex formation in the wake of an oscillating cylinder, J. Fluids Structures, 2

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