Effects of mass and damping ratios on VIV of a circular cylinder

ARTICLE IN PRESS Ocean Engineering 37 (2010) 511–519

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Effects of mass and damping ratios on VIV of a circular cylinder M.H. Bahmani, M.H. Akbari  Faculty of Mechanical Engineering, Shiraz University, Molla-Sadra Ave., Shiraz 71348-51154, Iran

a r t i c l e in fo

abstract

Article history: Received 5 February 2009 Accepted 12 January 2010 Available online 18 January 2010

In this study the basic characteristics of the dynamic response and vortex shedding from an elastically mounted circular cylinder in laminar flow is numerically investigated. The Reynolds number ranges from 80 to 160, a regime that is fully laminar. The governing equations of fluid flow are cast in terms of vorticity. The two-dimensional vorticity transport equation is solved using a vortex method. Effects of important parameters on the system response and vortex shedding are investigated; these include: mass ratio, damping ratio, Reynolds number and reduced velocity. The numerical results show that a decrease in either the mass ratio or damping ratio of the system can lead to an increase in both the oscillation amplitude and the reduced velocity range over which lock-in occurs. The results also suggest that the mass-damping parameter may characterize the system response adequately, although the effect of changing mass ratio appears to be a little more profound compared to damping ratio. Vorticity contour plots suggest that the vortex shedding occurs in the 2S mode, although a wake structure similar to the C(2S) mode appears at distances 15–20 diameters downstream in the lock-in region. The simulation results are in good agreement with previously published data. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Vortex-induced vibration Vortex method Damping ratio Mass ratio

1. Introduction Flow–structure interaction phenomena occur in many engineering fields. A fundamental flow–structure interaction problem is the flow-induced vibration of a structure due to vortex shedding. The case of vortex-induced vibration (VIV) of a circular cylinder has been given more attention partly because of its practical importance (such as in exchangers, marine cables and flexible risers in petroleum production). Vortex-induced vibration of an elastic structure is in general a nonlinear phenomenon. The vibration of the structure affects the flow field, which in turn changes the induced forces on the structure and hence the structural response. An elastically mounted circular cylinder exhibits a range of responses depending on a number of parameters including the reduced velocity, U* (U* = Ufn/D, where U, D and fn are flow velocity, cylinder diameter, and natural frequency of system, respectively), Reynolds number, Re, mass ratio m* (ratio of cylinder mass to displaced fluid mass, m* = (4m)/(prD2L), where m, r and L are the cylinder mass, fluid density and length of the cylinder, respectively), and damping ratio, x (ratio of damping coefficient to critical damping coefficient, z = c/ccritical). During vortex-induced vibration lock-in phenomenon may occur, in which case the vortex shedding and oscillation frequencies synchronize. This synchronization can lead to an amplification of cylinder’s vibrational response.

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E-mail address: [email protected] (M.H. Akbari). 0029-8018/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2010.01.004

The practical significance of VIV has led to a large number of studies. Several experimental investigations have been performed on VIV of a circular cylinder. Feng (1968) conducted VIV experiments in an air tunnel with a single one-degree of freedom flexible cylinder of large mass ratio and low damping (m*= 248, z = 0.00103, m*z = 0.255), at relatively large Re from 104 to 5  104. His work demonstrated that the cylinder experienced lock-in phenomenon in a range of reduced velocities of U* E5 8. The cylinder amplitude response in terms of reduced velocity was found to have two distinct branches. It was later shown by Brika and Laneville (1993) and Govardhan and Williamson (2000) that the initial branch is associated with the 2S mode of vortex formation, involving 2 single vortices shed per cycle, while the lower branch corresponds to the 2P mode, where 2 vortex pairs are shed per cycle (as originally defined by Williamson and Roshko, 1988). He also explored the effects of damping on both the amplitude and frequency response. Khalak and Williamson (1999) carried out VIV experiments using a setup with low mass–damping (m*z). They found that the system response, depending on the magnitude of mass–damping parameter (m*z), may be of two different types. For high m*z, they found two branches in their response, namely initial and lower branches, while for low m*z three modes of response were found, initial, upper, and lower branches. The upper branch is associated with high amplitude oscillation response. Anagnostopoulos and Bearman (1992) studied experimentally free vibration of circular cylinder in laminar flow (90 oRe o150). They observed that the oscillation amplitude was stabilized after many oscillation cycles within lock-in region. They also indicated

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that the maximum amplitude occurred near the lower limit of the lock-in region. This is in contrast to the results of some more recent numerical studies, e.g., Leontini et al. (2006) and Pasto (2008). Gharib (1999) has noted the absence of lock-in behavior from almost all experimental studies for small mass ratios. Jauvtis and Williamson (2003) studied the behavior of an elastically mounted cylinder with both streamwise and transverse motions. The range of m* in their study was between 5 and 25. It was found that the freedom of body to oscillate in both directions had very little effect on the transverse response of the system and the wake vortex dynamics. Anagnostopoulos (1994) investigated numerically the response and wake characteristics of a vortex excited cylinder in laminar flow. A finite element scheme was implemented for the numerical solution of the Navier–Stokes equations. His numerical results were found to be in good agreement with the experimental evidence except at reduced velocities above the lock-in region. Newman and Karniadakis (1996, 1997) performed a direct numerical simulation (DNS) study of the flow past a freely vibrating cable using body fitted coordinates at Reynolds numbers 100 and 200. Guilmineau and Queutey (2004) investigated VIV of a circular cylinder with low mass–damping in a turbulent flow using three sets of conditions: (a) flow starting from rest, (b) increasing velocity, and (c) decreasing velocity. In the first and third cases, the simulations predicted only the lower branch. On the other hand, with the increasing velocity condition, the maximum vibration amplitude matched with the corresponding experimental value, but the upper branch was not predicted in accordance with the experiments. Singh and Mittal (2005) simulated numerically VIV of a circular cylinder at low Reynolds numbers. They carried out two sets of computations to investigate the effects of Re and reduced natural frequency, Fn(Fn = 1/U*). They found that the effect of Re was very significant. Also, Prasanth and Mittal (2008) simulated numerically VIV of a circular cylinder of low mass ratio (m* =10) in laminar flow. They investigated different branches in cylinder response and different vortex shedding modes associated with each branch. They also studied the effect of blockage on system response and the hysteresis between the branches. They reported the C(2S) vortex shedding mode for high-amplitude oscillations in the lock-in region. Willden and Graham (2006) investigated vortex-induced vibrations of circular cylinders with zero damping at different mass ratios, using a two-dimensional numerical method. They found that at low Reynolds numbers Reo400, transversely vibrating circular cylinders exhibited three distinct regions of response, namely primary, secondary and tertiary response regions. The primary response is that normally associated with VIV and occurred around resonance. Secondary, or super-harmonically excited, response was found to occur for higher mass ratios considered (m*45), and the tertiary or infinite lock-in response was defined as a response that is maintained at an approximately constant amplitude for all reduced velocities above some threshold. Some researchers studied the effects of mass and damping on the system response (i.e., Khalak and Williamson, 1997; Klamo et al., 2006). Some of these studies suggest that the system response is characterized by combined mass–damping, m*z (Griffin, 1980; Naudascher and Rockwell, 1993); on the other hand some researchers state that the system response is influenced by the independent effects of mass and damping (Sarpkaya, 1995). Much research on this subject is discussed in the reviews by Williamson and Govardhan (2004, 2008). To date the majority of studies of VIV of a circular cylinder have been experimental and at Reynolds numbers where the flow is inherently turbulent and three dimensional. One rare laminar flow, free vibration experiment that is found in the literature is by Anagnostopoulos and Bearman (1992). A more recent numerical

study of this problem at Re=200 was performed by Leontini et al. (2006) using a spectral-element technique. In the present study we first simulated similar cases to those investigated by Anagnostopoulos and Bearman (1992) and Leontini et al. (2006), and examined the dynamic behavior of the system. We then simulated more cases to investigate the effects of mass and damping ratios on the system response, particularly the maximum oscillation amplitude and the extent of lock-in region. An outcome of such investigation is to verify whether the mass–damping parameter (m*z) can completely characterize the system response. It is known that VIV oscillation of a cylinder is a direct consequence of the periodic force applied to the system by the fluid. Therefore, an accurate prediction of the flow filed and the fluid force is essential in VIV simulations. Anagnostopoulos (1994) and Prasanth and Mittal (2008) implemented finite element schemes to solve the Navier–Stokes equations. However, a vortex method is used in the present study to simulate the flow field. This method is based on the discretization of the vorticity field into point vortices that are convected by the local velocity, and are diffused due to viscous effects of the fluid. Vortex methods are believed to be very suitable for separated flow fields such as those around bluff bodies (Ploumhans and Winckelmans, 2000; Lakkis and Ghoniem, 2009). One advantage of this method in our case is the relative ease in extending the simulation domain to hundreds of diameters from the cylinder without much increase in the computational time or memory. This domain is several times greater than that used by Anagnostopoulos (1994) in his computations. The computations by Prasanth and Mittal (2008) are made for cases with blockage in the flow field; such effect is not present in the present simulations.

2. Numerical approach The numerical algorithm developed in this investigation to simulate such unsteady flow field is based on a vortex method, described previously in detail by Akbari and Price (2003, 2005). The governing equations describing a two-dimensional incompressible flow may be expressed as the vorticity transport equation and the stream function-vorticity equation as @o ~ Þo ¼ nr2 o þ ð~ u :r @t

ð1Þ

r2 c ¼ o

ð2Þ

where o, c and ~ u are the vorticity, stream-function and velocity vector, respectively, and n is the kinematic viscosity of the fluid. The stream function and velocity components (u, v) are related by u¼

@c ; @y

v¼

@c @x

ð3Þ

Given the vorticity field, Eq. (2) can be solved for the stream function, and velocity components can then be calculated using the above relations. We used a vortex method to solve the vorticity equation, Eq. (1). In this vortex method the flow field is represented by point vortices, and the governing equations of the flow are solved by considering the creation, interaction and motion of these vortices. The mechanism for the creation of these vortices is provided by the no-slip condition on the solid boundary. An operator splitting method is used to solve Eq. (1). In this method, Eq. (1) is splitted into convection and diffusion parts as follows: @o ~ Þo ¼ ð~ u :r @t

ð4Þ

@o ¼ nr2 o @t

ð5Þ

ARTICLE IN PRESS M.H. Bahmani, M.H. Akbari / Ocean Engineering 37 (2010) 511–519

The time domain is discritized into small time-steps during which Eqs. (4) and (5) are solved sequentially. The vorticity in the interior field is provided by a thin layer of vortex sheet on the solid boundary via two mechanisms. One mechanism is the diffusion caused by the viscosity of the fluid, and the other is the convection through the bulk motion of the fluid. 2.1. Grid generation In the present vortex method, a computational grid is used for the following purposes: (a) solving the Poisson equation for the stream-function given the vorticity field, (b) calculating the velocity field given the stream-function, and (c) solving the diffusion equation for the vorticity field. New vortices are created at the location of the first row of the grid nodes, on the surface of the body, during each time step. We used a polar grid around the cylinder. The grid is uniform in the circumferential direction and stretched in the radial direction. The computational grid is fixed to the body, and hence moves with the cylinder as it vibrates. The computational domain is chosen large enough such that the flow on the outer boundary can be assumed to be uniform (diameter of the external boundary is 200 times larger than the cylinder diameter). Fig. 1 illustrates the entire grid and its details near the cylinder.

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2.2. Velocity field calculation At the start of each time steps the velocity field is calculated. To obtain the velocity field, the Poisson equation (Eq. (2)) is solved for the stream function using the vortex-in-cell method. The vorticity field is specified by the arrangement of the point vortices. An area-weighting distribution scheme (see Fig. 1(c)) is used to determine the nodal values of vorticity from the vorticity carried by the point vortices. Fig. 1(c) shows a typical cell of the grid, which in this case contains the jth point vortex with strength Gj located at ~ x j . The ray and the arc (concentric with the grid) that pass through ~ x j cut the cell into four segments. The surface areas of these segments are denoted as ai, i=1,y,4, such that ai corresponds to the segment on the opposite side of corner i, as shown in Fig. 1(c). The vorticity of the jth vortex is now distributed onto the four surrounding nodes as follows:

oi ¼

ai ðatotal Þ2

Gj ;

i ¼ 1; 2; 3; 4

ð6Þ

In this equation atotal is the total area of the cell, aj is described in Fig. 1(c), and Gj is the strength of the jth vortex. A uniform distribution of vorticity (Gj/atotal) over the cell due to the jth vortex is assumed. All other vortices are interpolated onto the grid

Fig. 1. Grid generated around the cylinder: (a) entire grid, (b) details of the grid near the body surface, and (c) typical polar cell containing a vortex.

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in a similar manner, and due to the linearity of the problem, the total vorticity at a node is simply the sum of the contributions from individual vortices. Now, given the discritized vorticity field, the Poisson equation (Eq. (2)) is solved for the stream function. For the boundary conditions for the Poisson equation, we assume the flow is uniform far away from the body,

c ¼ Uy

ð7Þ

where U is the free-stream velocity. On the surface of the vibrating body the fluid velocity should be equal to the instantaneous velocity of the cylinder:

c ¼ V cyl x

ð8Þ

Here Vcyl is the instantaneous cylinder velocity in the transverse direction, and x and y are the Cartesian coordinates of the node. A cyclic reduction method is used to solve the Poisson equation. This scheme is described in detail by Swartrauber and Sweet (1975), and is not presented here for the sake of brevity. 2.3. No-slip boundary condition The no-flow boundary condition is satisfied during the solution of the convection problem by specifying proper values for the stream function on the body surface, as described above. The no-slip boundary condition is satisfied by creating vortices at the solid boundary, such that the velocity field induced by these vortices cancels out the tangential component of velocity along the boundary. At the beginning of a time step the velocity field is calculated through the vortex-in-cell method. Thus, the tangential velocity component along the body surface is obtained. If the velocity component along the surface at node i is uti , and the length of the ith panel on the surface is si, a number of vortices with a total circulation (strength) of ðGi Þtotal ¼ uti si

ð9Þ

are created on panel i in order to cancel out the tangential velocity, uti . The total circulation (Gi)total is divided equally between n point vortices located at node i, such that the absolute value of the strength of each, Gj, does not exceed a given maximum value, Gmax. Creation of new vortices changes the vorticity filed and hence the velocity field. Therefore, at the end of each time step the vortexin-cell method is applied again to find the new velocity field in which both the no-flow and no-slip conditions have been satisfied. 2.4. Solution of the convection equation Eq. (4) states that the total derivative of vorticity is zero during convection, or that the vorticity of particles are conserved as they are convected. Therefore, to satisfy Eq. (4) the point vortices are convected with the local velocity for the period of a time step, once the velocity field has been calculated. The velocity field is thus computed after the creation of new vortices. Then the new positions of the point vortices are calculated according to the following equation: kþ1 k k ~ ¼~ xj þ ~ u ð~ x j ; tk Þ Dt xj k ~ xj

ð10Þ kþ1 ~ xj

is the position of the jth vortex at time tk, and is its where new position at time tk + 1 due to the convection part of the problem. 2.5. Solution of the diffusion equation An alternating-direction-implicit (ADI) finite difference scheme is used to solve the diffusion equation, Eq. (5). First the

vorticity of point vortices are distributed onto the mesh nodes. The diffusion equation is then solved by the ADI scheme, and the new nodal values of vorticity are obtained. Obviously, the nodal vorticity distributions before and after the solution of the diffusion equation are different. Therefore, the vorticity of point vortices must be updated. To do this, the nodal vorticity values are interpolated back to the point vortices as described below. Suppose that during the distribution of vorticity onto the grid, the jth vortex contributed an amount of vorticity to node p given by

$p;k j ¼

ap ðatotal Þ2

Gkj

ð11Þ

where Gkj is the strength of the jth vortex before the diffusion process (at time tk), ap is the surface area of the segment (of the cell) that corresponds to nod p, and at is the total area of the cell containing the jth vortex. Also suppose that at the end of vorticity distribution the total vorticity at node p was okp . Now, if after solving the diffusion equation (at time tk + 1) the vorticity at node p is okp þ 1 , then the contribution to the jth vortex from node p is þ1 $p;k ¼ j

$kp þ 1 p;k $j okp

ð12Þ

This is the contribution to the jth vortex from node p only. The total strength of this point vortex after the diffusion process, will be the sum of the contributions from all four surrounding nodes, 4

þ1 Gkj þ 1 ¼ ð S $n;k Þðatotal Þj j n¼1

ð13Þ

After distributing the nodal vorticity among the existing vortices, if a node still contains any remaining vorticity, new vortices are created at the location of that node with a total strength of

Gtotal ¼

ðokp þ 1 Þleftover atotal

ð14Þ

where ðopk þ 1 Þleftover is the amount of the remaining vorticity at node p, and atotal is the average corrected surface area of the four surrounding cells. The condition |Gj| o|Gmax| applies here in the same manner as applied in the creation of new vortices, thus if |Gtotal| 4|Gmax| more than one vortex is created. 2.6. Force calculation The force acting on the rigid boundary is due to both the pressure and shear stress distributions on its surface. 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 integrated, with pressure at the outer limit of the mesh taken constant (zero) everywhere. The shear stress at any point on the body surface is approximated by a finite difference scheme as well. Given the pressure and shear stress distributions, the lift and drag forces can be easily obtained, and expressed in non-dimensional forms as Cl = 2Fl/(rU2D) and Cd = 2Fd/(rU2D), where Fl and Fd are the calculated lift and drag forces, respectively. 2.7. Structural dynamics At the end of each time step, once the fluid forces on the cylinder have been calculated, the equation of motion of the cylinder in transverse direction is solved to find its new location and transverse velocity. The transverse displacement of the cylinder, y(t), due to the fluid force loading per unit span, Fy(t), is modeled using a mass, spring and damper model. Thus, the

ARTICLE IN PRESS M.H. Bahmani, M.H. Akbari / Ocean Engineering 37 (2010) 511–519

cylinder’s equation of transverse motion is given by my€ þcy_ þ ky ¼ Fy

ð15Þ

where m is the cylinder mass, c the damping coefficient, and k the structural stiffness. A fourth-order Runge–Kutta scheme is used to solve Eq. (15). A curve fitting on the lift force during previous time steps is used to estimate its value in the new time step.

3. Results In this work, vortex-induced vibration of an elastically supported circular cylinder at low Reynolds numbers is numerically investigated. Although the main purpose is to study the VIV of a cylinder, but to examine the accuracy of the solution method and the CFD code, the flow past a fixed circular cylinder at Reynolds numbers in the range of 70 oRe o180 was first investigated. The calculated mean drag coefficient,C d , lift coefficient amplitude, Cl0 , and the Strouhal number, St (St= fvU/D, where fv is vortex shedding frequency) for these cases were compared with experimental or numerical results reported earlier, and very good agreements were obtained. The present numerical results for Re = 100, for example, are compared in Table 1 with available data where an acceptable agreement is observed. The variation of St with 1/ORe is compared to the experimental results of Fey et al. (1998) and the correlation proposed by Williamson and Brown (1998) in Fig. 2. In this case also a very good agreement is observed. The agreement between the present numerical results and previous experimental or numerical data, points to the accuracy of the solution method implemented in our simulations. To date the majority of investigations of VIV of a cylinder have been at Reynolds numbers where the flow is inherently turbulent and three dimensional. An experimental study of free vibrations of a cylinder in a laminar flow was performed by Anagnostopoulos and Bearman (1992). Also, a more recent numerical investigation of this problem was also presented by Leontini et al. (2006). It was Table 1 Comparison of results for a fixed cylinder at Re = 100. Reference

Cl0

Cd

St

Stansby and Slaouti (1993) Anagnostopoulos (1994) Handerson (1995) Shiels et al. (2001) Present study

0.35 0.27 0.33 0.33 0.33

1.32 1.20 1.35 1.33 1.33

0.166 0.167 0.166 0.167 0.165

0.2 0.18

St

0.16 0.14 0.12 0.1 0.07

0.08

0.09

0.1

0.11

0.12

1/√Re Fig. 2. Variation of St with 1/ORe for a fixed cylinder: (’) present numerical results; (—) experimental correlation of Fey et al. (1998) and (—) St–Re correlation of Williamson and Brown (1998).

515

therefore decided to simulate similar cases in order to validate our simulation code. Consequently, the system response is investigated with variations in mass and damping ratios. At the start of each simulation the cylinder is held fixed at its equilibrium position until a periodic flow is established. For each reduced velocity the time history of the cylinder displacement and the lift force are recorded until the cylinder oscillation amplitude settles to a steady state. Applying a power spectrum diagram (PSD) analysis, vortex shedding and cylinder oscillation frequencies are obtained. A comparison of these frequencies is made to check for lock-in.

3.1. Validation The system parameters are first set according to Anagnostopoulos and Bearman (1992) experiments as: m* =149, fn =7.016, z = 0.0012. The reduced velocity is varied between 5 and 9. The change in reduced velocity is achieved by altering the flow velocity, hence the Reynolds number will also change according to Re = 17.9U*; therefore the Reynolds number range is 80–160 in these simulations. The time traces of the cylinder displacement and lift force were obtained for each case, but are not presented here for the sake of brevity. To find the lock-in region the vortex shedding frequency was obtained in each case (by PSD analysis of the lift force) and compared to the natural oscillation frequency. It was seen that outside of the lock-in region the vibration amplitude of the cylinder is very small. A beating effect was also observed in such cases, which is in agreement with the experimental results of Anagnostopoulos and Bearman (1992). Increasing the reduced velocity beyond U* = 5.75 (Re= 103) up to U* =7.26 (Re =130), results in a sudden change in the system response. In this case, the oscillation amplitude increases gradually before it reaches a steady value after many oscillation cycles; it is also noticeable that the vortex shedding frequency diverges from that for a stationary cylinder and synchronizes with the cylinder oscillation frequency. Variation in the system response with increasing reduced velocity (U*) was investigated in this manner, and the results are summarized in Fig. 3. In this figure, the variation of the maximum amplitude with U* (or Re) are presented and compared with both the experimental results of Anagnostopoulos and Bearman (1992) and the numerical results of Anagnostopoulos (1994). Although there are some differences between our simulation results and the previously published data, the general trend of the lock-in region is captured well. The maximum amplitude was found near the lower limit of the lock-in region by Anagnostopoulos and Bearman (1992), while in the present numerical study the maximum amplitude is found somewhere in the first half of the lock-in region. However, this is in agreement with the results of Leontini et al. (2006) and Pasto (2008) for laminar flows and similar m*z values. Nevertheless, the reduced velocity at the onset of lock-in and the reduced velocity band over which lock-in occurs are in agreement with the experimental data of Anagnostopoulos and Bearman (1992). In the same way, simulations were performed for cases investigated by Leontini et al. (2006) in order to examine the amplitude of the system response as a function of the reduced velocity (U*). The system parameters are set accordingly as: Re =200, m* =10, z =0.01, with the reduced velocity varied between 4 and 7. A comparison of the results are presented in Fig. 4, where a close agreement is observed between the present prediction for the amplitude of response and that given previously by Leontini et al. (2006). It is seen that in both sets of the numerical data, the maximum amplitude of response

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Fig. 3. Variation of the non-dimensional oscillation amplitude with U* (Re): (’) present numerical results; (J) experimental results (Anagnostopoulos and Bearman, 1992) and (+ ) previous numerical results (Anagnostopoulos, 1994).

Table 2 Variation of the maximum amplitude and synchronization zone with m* and z.

0.6

Case

0.5 0.4 A/D

Fig. 5. Variation of the non-dimensional oscillation amplitude with Re for three different mass ratios.

1 2 3 4 5

0.3 0.2

m*

149 298 74.5 149 149

z

0.0012 0.0012 0.0012 0.0024 0.0006

m*z

0.1788 0.3576 0.0894 0.3576 0.0894

A*max

0.48 0.36 0.52 0.40 0.49

Synchronization zone U*

Re

5.75–7.26 5.98–6.99 5.76–7.55 5.98–7.26 5.76–7.65

103–130 107–125 103–135 107–130 103–137

0.1 0 4

4.4

4.8

5.2

5.6

6

6.4

6.8

U* Fig. 4. Variation of the non-dimensional oscillation amplitude with U*: (’) present numerical results and (J) previous numerical results (Leontini et al., 2006).

has been predicted to occur somewhere in the first half of the lock-in region. This close agreement gives us confidence in the accuracy of the simulations performed by the present numerical model.

3.2. Effect of mass ratio Vortex-induced vibration of a circular cylinder is in general a nonlinear phenomenon and the system response depends on a number of parameters including the reduced velocity, U*, Reynolds number, Re, mass ratio, m* and damping ratio z. We now study variation of the system behavior due to changes in mass and damping ratios. Three mass ratios of m* = 74.5, 149, 298 are considered, while the damping ratio is fixed at z =0.0012. To see the effect of mass ratio more clearly, variations of stable oscillation amplitude versus reduced velocity are plotted in Fig. 5 with the mass ratio as a parameter. It is seen in this figure, for example, that at U* = 5.87 (Re= 105) the amplitude of oscillation is considerable for cases with m* = 74.5 and 149, while in the case with m* =298 the oscillation amplitude is small. By comparing the cylinder oscillation frequency with the vortex shedding

frequency, it is found that in the cases with m* = 74.5 and 149 lock-in has occurred, while in the case with m* = 298 lock-in has not occurred. It is also observed that changing the mass ratio not only alters the amplitude response of the cylinder, it also changes the extent of lock-in region. The non-dimensional maximum amplitude (A*max =Amax/D) and the extent of the synchronization zone are summarized in Table 2 for different value of the mass and damping ratios. Cases 1, 2 and 3 in this table correspond to the variation of the system response with mass ratio. These results indicate that decreasing mass ratio causes an increase in the oscillation amplitude and an increase in the reduced velocity band over which lock-in occurs. Referring to Fig. 5 and Table 2, one can observe that the maximum oscillation amplitude for the case with m* = 74.5 is about 0.52D. Once the mass-ratio is doubled to m* =149, the maximum amplitude slightly decreases to 0.48D. However, when the mass ratio is doubled again (m* = 298), a considerable change in the oscillation amplitude is observed (see Fig. 5 and Table 2). The maximum oscillation amplitude in this case is decreased to 0.36D. These results indicate that the behavior of the system with respect to mass-ratio is in general non-linear.

3.3. Effect of damping ratio Three damping ratios of z =0.0024, 0.0012, 0.0006 are considered now, while the mass ratio is fixed at m* = 149. The maximum amplitude of oscillation versus reduced velocity is plotted in Fig. 6 for these three damping ratios. It is observed that the damping ratio, similar to the mass ratio, controls both the

ARTICLE IN PRESS M.H. Bahmani, M.H. Akbari / Ocean Engineering 37 (2010) 511–519

Fig. 6. Variation of the non-dimensional oscillation amplitude with Re for three different damping ratios.

maximum amplitude and lock-in region of the system. At U* = 5.87 (Re =105), for example, the oscillation amplitude is considerable with z = 0.0006 and 0.0012, while in the case with z = 0.0024 the amplitude of oscillation is small. By comparing the cylinder oscillation frequency with the vortex shedding frequency, it is also found that in the cases with z = 0.0006 and 0.0012 lock-in has occurred, while in the case with z = 0.0024 it has not. The maximum amplitude of oscillation and the synchronization band of reduced velocity for different damping ratios and a fixed mass ratio are presented in Table 2 under cases 1, 4 and 5. Considering these data and the results shown in Fig. 6, it is observed that increasing the damping ratio decreases both the maximum oscillation amplitude and the velocity range over which lock-in occurs. One can also realize that the system response with respect to the damping ratio is non-linear. 3.4. On mass–damping parameter (m*z) It has been more than three decades that many researchers have attempted to determine the extent at which a combined mass–damping parameter (m*z) could characterize VIV response of a system. Skop and Griffin (1973) introduced a parameter (which was later symbolized as SG, and called Skop–Griffin number), and proposed that the maximum amplitude of oscillation at all reduced frequencies (A*max = Amax/D) is a function of this parameter only. Skop–Griffin number (SG) was defined as SG ¼ 2p3 ðStÞ2 ðm zÞ

ð16Þ

In this equation, St is the Strouhal number of a stationary cylinder. Later Griffin (1980) used a log–log plot of A*max versus SG; this type of plot is known as a Griffin plot. Khalak and Williamson (1999) modified the Griffin plot and plotted A*max versus (m* +CA)z, where CA is the potential added mass coefficient. On the other hand some researchers Sarpkaya (1978, 1995) firmly stated that the response is governed by m* and z independently and not by the combined mass–damping m*z. In the present study, five cases were studied with different mass and damping ratios (Table 2). Considering the case with m* = 149 and z = 0.0012 as the base case, one can double m*z by either doubling m* (to 298, case 2) or doubling z (to 0.0024, case 4). Comparing the results of these two cases (cases 2 and 4 in Table 2) with the base case (case 1), one can observe that increasing m*z (by either doubling m* or z) will decrease both the

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Fig. 7. Griffin plot (Scruton plot) of the amplitude response: (’) present results (different cases according to Table 1); (&) (Newman and Karniadakis, 1997); (J) Anagnostopoulos and Bearman (1992) and (—) curve fit of Sarpkaya (1978).

maximum amplitude and the velocity range over which lock-in occurs. The same result can be deduced by comparing cases 3 and 5 in Table 2 or Figs. 4 and 5 with each other. A Griffin plot (or a Scruton plot, as suggested by Zdravkovich (1982) of the present amplitude data is shown in Fig. 7, where a comparison is also made with previously available data. Although the pair cases 2 and 4 or and 5 do not fall exactly on the same points in the graph, however, variations in m*z (by either doubling or halving m* or z) influence the system response (in terms of A*max) and the general trend of previous results are followed by the present data to an acceptable extent. The above observations suggest that mass ratio and damping ratio have similar influences on the system response. In fact, the system response appears to be characterized by combined mass– damping parameter (m*z) with acceptable accuracy, similar to what was proposed by Griffin (1980) or Khalak and Williamson (1999). However, more recent studies by Blevins and Coughran (2009) suggest a correlation between combined mass–damping parameter and maximum vortex-induced vibration response for transverse vibration but not for two-dimensional motion at large amplitudes.

3.5. The vorticity field Different vortex formation modes can be induced by cylinder motion, such as 2S, 2P, S+ P. Williamson and Roshko (1988) studied experimentally vortex formation in the wake of a forced vibrating cylinder. They described other patterns such as those formed by coalescence of vortices. It was also shown that the 2P mode appears only when the vortex shedding becomes turbulent (Re4200). Brika and Laneville (1993) were the first that observed the 2P mode in free vibration of a cylinder in turbulent flow. Anagnostopoulos and Bearman (1992) observed only the 2S mode in their low Reynolds number experiment. The S+ P mode is observed in some forced vibration studies (Griffin and Ramberg, 1974; Williamson and Roshko, 1988). Williamson and Govardhan (2004) believe that an S+P mode is not able to excite body into free vibration. Shown in Fig. 8 are the instantaneous vorticity fields at four selected reduced velocities (Reynolds numbers) for a case with m* =149 and z = 0.0012 as predicted in our simulations. For all the Reynolds number range considered here, vortex shedding is of 2S

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acceptable agreement between the two sets of results was obtained. The system response due to changes in mass and damping ratios was then studied. It was found that for this range of Re and mass and damping ratios, variations of mass ratio or damping ratio have almost the same result on the system response. Increasing m*z (by either varying m* or z) will decrease the maximum amplitude and the velocity range over which lockin occurs. The influence of the combined mass–damping parameter was also examined, and it was observed that although the influence of mass ratio on the system response is a little more significant than damping ratio, the present results follow the general trend of the Giffin plot as suggested by Griffin (1980) or Khalak and Williamson (1999). In other words, it appears that the system response is characterized by combined mass–damping parameter (m*z) with acceptable accuracy. Finally, examining the vorticity contours in the wake of the free vibrating cylinder in the synchronization regime, the 2S vortex shedding mode was observed at low Reynolds numbers; this is in contrast to the 2P mode of vortex shedding which is observed in the turbulent flow regime during lock-in.

References

Fig. 8. Instantaneous vorticity contours for a free-vibrating circular cylinder with m* = 149 and z = 0.0012 at different reduced velocities (Reynolds numbers): (a) U* = 5.03 (Re = 90); (b) U* =5.87 (Re =105); (c) U* = 6.13 (Re = 110); and (d) U* = 8.39 (Re = 150).

type (two oppositely vortices shed per cycle). Parts (a) and (d) of the figure show the vorticity contours for Re = 100 and 150, respectively, where lock-in did not occur. The vorticity contours in these cases are very similar to those for a fixed cylinder at the same Reynolds numbers. Parts (b) and (c) of the figure show vorticity contours for two cases in which lock-in occurs and the cylinder experiences a high amplitude oscillation. Comparing the vorticity patterns shown in Fig. 8(b) and (c) with those for a fixed cylinder at the same Re, it is observed that although the vortex shedding mode is still 2S, the vorticity patterns are somehow different from the standard 2S. It is seen that vortex spacing is smaller in these cases, and that the vortices in the far wake coalesce at distances 15–20 diameters downstream shifting towards the C(2S) mode of vortex shedding (as described by Williamson and Roshko, 1988). The C(2S) is very similar to the 2S mode except that vortices coalesce in the wake. This is similar to the two-dimensional numerical results of Singh and Mittal (2005) and Prasanth and Mittal (2008) who have also reported C(2S) shedding at high-amplitude oscillations in the lock-in region at low-Re flows.

4. Summary and conclusion Flow-induced vibration of a low-damped, elastically mounted circular cylinder in laminar flow was simulated using a twodimensional vortex method. The main purpose in this work was to investigate the system response due to changes in mass and damping ratios in the low-Reynolds number regime. First, a case was simulated for which experimental results were available. An

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