Vortex-induced vibration (VIV) of two elastically coupled side-by-side circular cylinders in steady flow

Kalyani Kaja, Ming Zhao and Yang Xiang

School of Computing, Engineering and Mathematics, University of Western Sydney, Locked Bag 1797, Penrith, NSW 2751, Australia

k.kaja@uws.edu.au

ABSTRACT

Vortex-induced vibration (VIV) of two elastically coupled side-by-side circular cylinders in steady flow at a low Reynolds number of 150 and mass ratio of 2 is investigated numerically. The focus of the study is to investigate the influence of the interference between the two circular cylinders on the response amplitude and frequency. The VIVs of both cylinders are confined in the cross-flow direction only. Simulations are carried out for the reduced velocities ranging from 1 to 30 to ensure that the whole lock-in regime is covered. The vibration of the two cylinders may lock-in with one of the two natural frequencies, depending on the reduced velocity. It is found that the lock-regimes of the two natural frequencies overlap, forming a wider lock-in regime that is wider than that of a single cylinder. The response amplitude reaches it maximum when the vibration frequency is close to either the first mode or the second mode natural frequency.

Key words: Vortex shedding; vortex-induced vibration; circular cylinder

1. INTRODUCTION

Vortex-Induced vibration (VIV) of cylindrical structures is of practical interest to many fields of engineering and has attracted much attention from numerous researchers in the past few decades. It is well known that when a single elastically mounted cylinder is placed in a free stream, large-amplitude oscillations occur when the shedding frequency synchronizes with the oscillation frequency Williamson & Roshko (1988). Numerous experiments have shown that when the vortex shedding frequency is the same as the natural frequency, the natural frequency of an elastically mounted rigid cylinder takes control of the vortex shedding in apparent violation of Strouhal relationship. Then the frequency of vortex shedding and the oscillation frequency of the cylinder collapse into a single frequency, which is known as the lock-in phenomenon Sarpakaya and Isaacson (1981). Unlike forced vibrations of an oscillator, the synchronization occurs over a range of velocities, called the ‘lock-in’ or ‘synchronization’ region (Sarpkaya 2004; Williamson & Govardhan 2004). Feng (1968) conducted the well-known experiments on the one-degree-of-freedom (1DOF) vibration of a circular cylinder in the air flow. In his study the typical lock-in phenomenon was presented. In the lock-in regime, the vibration frequency of the cylinder is locked on to the natural frequency. In general, when a cylinder is exposed to a flow at a high mass ratios (the ratio of the mass of the cylinder to the mass of the displaced fluid), only two amplitude

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response branches exist, i.e., initial branch and lower branch. If the cylinder is placed in a flow at low mass ratios, the third branch (i.e., upper branch) is observed. Khalak and Williamson (1996; 1999) and Brika and Laneville (1993) concluded that the jump of vibration amplitude from the initial branch to the lower branch corresponds to a mode change from 2S to 2P. Herein, 2S stand for “two single vortices formed per vibration cycle” and 2P stands for “two pair of vortices are shed from the cylinder per vibration cycle”. Jauvtis and Williamson (2004) and Blevins and Coughran (2009) studied two-degree-of-freedom (2DOF) VIV of a circular cylinder at low mass ratios and observed a new response branch the super upper branch. The vibration amplitude in the super upper branch was 1.5 times of the cylinder diameter. Vibration of a circular cylinder in steady flow close to a plane boundary is also investigated due to its engineering importance (Fredsøe et al., 1985; Gao et al., 2006; Yang et al., 2006, 2008); Zhao and Cheng (2011). It was found that the plane boundary had significant effects on the vibration. Numerical simulations have been a powerful tool for studying VIV of cylinders. The numerical models based on the Navier-Stokes equations have provided satisfactory numerical results of VIV of a circular cylinder at low mass ratios. A number of recent studies on VIV were focused on low Reynolds numbers. Mittal & Kumar (2001) carried out two-dimensional simulations for two elastically mounted cylinders in tandem for Re = 100 and L/D = 5.5. Jester & Kallinderis (2004) simulated two-dimensional free vibrations for the tandem configuration for Re = 1000 and L/D = 5. For such spacing of L/D = 5 the flow will fall into the wake interference region as classified by Zdravkovich & Pridden (1977). Therefore, the upstream cylinder behaves as an isolated single cylinder, while the downstream cylinder experiences large flow-induced vibration over a wide range of flow velocities. Mittal and Kumar (1999) discovered the so-called “soft-lock-in” phenomenon where the vortex-shedding frequency of the oscillating cylinder did not exactly match the natural frequency through a numerical simulation of 2DOF VIV of a circular cylinder at a Reynolds number of 325. Prasanth et al. (2006) studied the effects of the blockage on the VIV of a circular cylinder for Reynolds numbers less than 150 numerically. Li et al. (2011) captured the nonlinear phenomenon of lock-in, beat and phase-switch (the phase difference between the lift force and the displacement in the cross-flow direction changing from 0° to 180°) by simulating the VIV of a circular cylinder at Re =200 numerically. Ji et al. (2011) studied the VIV modes of a circular cylinder at low Reynolds numbers and the effects of Skop–Griffin parameter (SG) on the cylinder response. In this study, numerical simulations are carried out to investigate the VIV of Vortex-induced vibration (VIV) of two side-by-side elastically coupled circular cylinders in steady flow at a low Reynolds number of 150. The focus of the study is to investigate the influence of the interference between the two circular cylinders on the response amplitude and frequency. The vibration of the two cylinders may lock-in with one of the two natural frequencies, depending on the reduced velocity. Simulations are carried out for the reduced velocities ranging from 1 to 30 to ensure that the whole lock-in regime is covered.

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2. NUMERICAL METHOD

The two side-by-side elastically coupled circular cylinders are considered in this study. The cylinders are placed in steady flow and allowed to only vibrate in the cross flow direction of the flow. The two cylinders are having the same diameter and the same mass. The stiffness of the springs is K. The vibration system is of two-degree-of-freedom. In this study, the distance between the two cylinder centres is a constant of L=3D with D being the diameter of the cylinders. The differential equation of motion governing the vibration of the two-degree-of-freedom of system is

MX+CY+KY=Fy (1)

where M, C and K are the symmetric mass, damping and stiffness matrices,

respectively. The mass matrices is M= 00

for both the cylinders. The stiffness

matrices K= 22

for the two cylinders. The hydrodynamic forces Fy can be

calculated by integrating the pressure and the shear stress over the cylinder surfaces. 2.1 Numerical model: The flow around the two side-by-side elastically coupled circular cylinders is simulated numerically. Vortex-induced vibration (VIV) of two side-by-side elastically coupled circular cylinders in steady flow as shown in Fig.1 is considered.

Fig. 1 Sketch of two elastically coupled side by side cylinders in fluid flow

The governing equations for simulating the laminar flow are the unsteady two-dimensional incompressible Navier-Stokes equations. In this study, the Arbitrary Lagrangian Eulerian (ALE) scheme is applied to deal with the moving boundaries of the cylinder. In the ALE scheme, the nodes of the computational mesh are allowed to move independently of the fluid velocity in order to avoid excessive mesh distortion. For the

Flow

Cylinder1

Cylinder2

L

y

x

k

k

k

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uniform flow in the cylinder the inlet boundary conditions are set as u=Vr, v=0.The velocity, length, time and pressure are non-dimensionlized by Zhao and Cheng (2012)

Df

uu

n

ii

1

,

D

xx i

i

, tft n 1 ,

221Df

pp

n

(2)

where x1 = x and x2 = y are the Cartesian coordinates in the in-line and transverse directions of the flow respectively, D is the diameter of the cylinders, ui is the fluid velocity component in the xi direction, p is the pressure, ρ is the fluid density, ν is the kinematic viscosity and fn1 is the first node structural natural frequency of the cylinder system. The primes in Eq. (2) stand for the dimensional variables. The nondimensional incompressible Navier-Stokes equations can be expressed as

0

i

i

x

u, (3)

2

2

Reˆ

i

ir

ij

ijj

i

x

uV

x

p

x

uuu

t

u

(4)

where iu is the velocity of the mesh movement, Re is the Reynolds number based on

the diameter of the cylinder D, the approach flow velocity U and the kinematic viscosity of the fluid ν i.e. Re=UD/ν. Vr is the reduced velocity defined by Vr=(U)/(fn1D).The boundary conditions for the governing equations are given for solving the Navier Stokes equations. At the surfaces of the cylinders, the no-slip boundary condition is employed, i.e. the fluid velocity on each cylinder surface is same as the vibrating speed of the cylinder. The specific dissipation rate ω is given at the nodal points next to the wall surface as 2

1Re/6 , where ∆1 is the distance from the wall. The inlet velocity

boundary conditions are set as rVu , 0v . At the outflow boundary, the gradients of fluid velocity in the direction normal to the boundary are set to be zero. Pressure at the outflow boundary is given a reference value of zero. After each computational time step, the boundary of the computational domain changes because of the displacement of the cylinders. The positions of finite element nodes are moved accordingly by solving the modified Laplace equation Zhao and Cheng (2011)

0 iS , (5)

where, iS represents the displacement of the nodal points in the ix direction, γ is a

parameter that controls the mesh deformation. In order to avoid excessive deformation of the near-wall elements, the parameter γ in an finite element is set to be A/1 , with A being the area of the element. The displacement of the mesh nodes is the same as the displacement of the cylinder on the cylinder surface and zero on other boundaries. By giving the displacements at all the boundaries, Eq. (5) is solved by a Galerkin FEM. The displacement of the mesh at the surface of each cylinder is the same as the displacement of the cylinder. At inlet, out and the two side boundaries, the displacement of the mesh is zero. 3. NUMERICAL RESULTS

The study is focused on the influence of the interference between the two circular cylinders on the response amplitude and frequency. The cylinder mass ratio m* is

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constant which is 2. The damping factor of cylinder is 0 in all numerical simulations. The initial gap between the two cylinders is 3D where D is the diameter of the cylinders. The Reynolds number is fixed to be 150 and the structural damping ratio is zero in all the numerical simulations. Simulations are carried for the reduced velocities ranging from 1 to 30 with an interval of 1. The reduced velocity for the two cylinder system is defined based on the first mode structural natural frequency as )/( 1DfUV nr . The

nondimensional two natural frequencies of the cylinders (fn1, fn2) is i.e., fn1=1 and fn2=1.732. The ratio of the natural frequency is fn2/fn1=1.732 where fn1 and fn2 are the first mode and second mode natural frequencies respectively. The numerical results of VIV of the two cylinder system are compared with those of 1DOF of vibration of a circular cylinder in the cross-flow direction. A rectangular computational domain with a width in the flow direction of 40D and a height in the cross-flow direction of 60D is used. The computational domain is divided into 17398 quadrilateral linear finite elements. Fig. 2 shows the computational mesh around the two cylinders. Refined elements are used close to the cylinder surface in order to capture the big variations of the flow field. The minimum mesh size at the cylinder surface is 0.0006D.

Fig. 2 Computational mesh around the cylinders

Fig. 3 shows the time histories of the displacements of the two cylinders at some typical reduced velocities in the lock-in regime. At reduced velocity of Vr=2, 3 and 4 the cylinder vibrates periodically with respect to time. At Vr=4, the vibration changes from one mode to another at about t=330. It can be seen that the time history of each cylinder before t=300 is the mirror image of that after t=330 with respect to the Y=0 line. At the reduced velocity of Vr=5 the vibrations of both cylinder are irregular. At Vr=6 the cylinders vibrate regularly at some period of time and then becomes irregular repeatedly. It appears that the vibration changes from one mode where the modes irregularly at Vr=6. Sometimes the vibration amplitude of one cylinder is greater than another and sometimes the vibration amplitude of the two cylinders are close to each other. At Vr=7 the vibration of cylinder is regular and periodic vibration. The vibrations of the two cylinders are perfectly periodic for all the reduced velocities greater than 7.

Y

-10 -5 0 5 10-10

-5

0

5

10

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As Vr>07, the distance between two mean position of the two cylinders increase with an increasing in Vr due to the increase in the repulsive force between the two cylinders. It can be seen in Fig. 3 that the vibrations of the two cylinders are in anti-phase with each other as the reduced velocity is 2, 5 and 6 and in-phase with each other for the rest of the reduced velocities.

Y

300 305 310 315 320 325 330 335 340 345 350 355 360-0.02

-0.01

0

0.01

0.02Cylinder 1Cylinder 2

(a) Vr=02

Y

300 305 310 315 320 325 330 335 340 345 350 355 360-1

-0.5

0

0.5

1Cylinder 1Cylinder 2

(b) Vr=04

Y

300 305 310 315 320 325 330 335 340 345 350 355 360-1

-0.5

0

0.5

1Cylinder 1Cylinder 2

(c) Vr = 05

Y

300 305 310 315 320 325 330 335 340 345 350 355 360-1

-0.5

0

0.5

1Cylinder 1Cylinder 2

(d) Vr = 06

Y

300 305 310 315 320 325 330 335 340 345 350-1

-0.5

0

0.5

1Cylinder 1Cylinder 2

(e) Vr = 07

Y

50 55 60 65 70 75 80 85 90 95 100-0.8-0.6-0.4-0.2

00.20.40.60.8

Cylinder 1Cylinder 2

(f) Vr = 10

Y

50 55 60 65 70 75 80 85 90 95 100-0.8-0.6-0.4-0.2

00.20.40.60.8

Cylinder 1Cylinder 2

(g) Vr = 11

Y

50 55 60 65 70 75 80 85 90 95 100-0.6

-0.4

-0.2

0

0.2

0.4

0.6Cylinder 1Cylinder 2(h) Vr = 12

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Fig.3 Time histories of the displacements of the cylinders

Fig. 4 shows the variations of the response frequency of the two cylinders with the reduced velocity. Fig. 5 shows the variations of the response amplitude of the two cylinders with the reduced velocity. The frequency and amplitude of VIV of a single cylinder is also plotted in Fig.4 and Fig.5 respectively. For a single cylinder, the response amplitude starts increasing with the increasing reduced velocity at Vr=3 and reaches its maximum at Vr=4. The response at Vr=3 and 4 is in the initial branch (Khalak and Williamson, 1996). In the reduced velocity range of 85 rV , the response frequency of a single cylinder is locked onto the natural frequency of the cylinder (i.e. the nondimensional frequency is 1). The amplitude of a single cylinder in the range of

85 rV decreases with an increase in reduced velocity. As the reduced velocity exceeds 9, the response of a single cylinder has been outside the lock-in regime, where the response amplitude is very small and the response frequency increases linearly with the increasing reduced velocity. Now the response of the two cylinder system will be discussed. It can be seen in Fig. 4 and Fig. 5 that, because of the symmetric configuration, both the response frequency and the response amplitude of cylinder 1 are the same as their counterparts of cylinder 2, respectively. The variation of the response frequency with the reduced velocity in the two cylinder system is similar to that in the single cylinder system. The frequency generally increases with increasing reduced velocity. It is expected that, similar to that of a single cylinder, the response of the two cylinders may locks onto the first mode frequency or the second mode frequency, depending on the reduced velocity. At Vr= 4, the response frequency approaches to 1 and the response amplitudes reaches its maximum. The response amplitudes at Vr=5 and 6 are almost the same as that of a single cylinder. Another maximum value of the response amplitude occurs at Vr=7, where the response frequency is very close to the second mode natural frequency. The response frequency follows the linear function of the reduced except in the range of in the reduced velocity range of 123 rV . The reduced

range of 123 rV is believed to the lock-in regime because the vortex shedding frequency is found to be the same as the vortex shedding frequency and the response amplitude is significant greater than that outside this regime. Govardhan and Williamson (2000) found that the vibration frequency increases with the increasing reduced velocity gradually in the upper branch until at the boundary between the upper branch and the lower branch, where the response frequency keeps constant. In this study, the response frequency increases slowly with the reduced velocity in the range

Y

50 55 60 65 70 75 80 85 90 95 100-0.4

-0.2

0

0.2

0.4Cylinder 1Cylinder 2

(i) Vr = 18

Y

50 55 60 65 70 75 80 85 90 95 100-0.4

-0.2

0

0.2

0.4Cylinder 1Cylinder 2

(j) Vr = 20

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of 125 rV . However, lower branch where the response frequency is constant is not observed.

Fig. 4 Variations of response frequencies with the reduced velocity

Fig. 5 Variations of the response ampliutdes with the reduced velocity

Fig. 6 shows the spectra of cylinder 1 in the two-cylinder system, which are obtained based on the FFT (Faster Fourier Transform). The spectrum of cylinder 2 in each reduced velocity is the same as that of cylinder 1 and is not shown in Fig. 6. It can be seen in Fig. 6 that, although the vibration is irregular, a well defined peak frequency can be clearly identified in each of the spectra for Vr=5 and 6. The spectrum for any reduced velocity greater than 7 is dominated by a single peak as shown in Fig. 6 (d).

0

1

2

3

4

5

6

0 5 10 15 20 25 30

f

Vr

Single-cylinder Cylinder-1 Cylinder-2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30

A

Vr

Single-cylinder Cylinder-1 Cylinder-2

f

Ay

0 1 2 30

0.1

0.2

0.3

0.4

0.5 a) Vr=04

f

Ay

0 1 2 30

0.1

0.2

0.3

0.4

0.5

b) Vr=05

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Fig. 6 Amplitude spectra of the response of cylinder1 in the two cylinder system

The vortex shedding flow structure is represented by the vorticity contour. The vorticity is defined as the curl of the fluid's velocity. A vortex is a region within a fluid where the flow is mostly a spinning motion about an imaginary axis, straight or curved. Fig. 7 shows the vorticity contours around the two cylinders at some typical reduced velocities in the lock-in regime. At Vr=2 the cylinder vibrates and two vortices are shed from each of the cylinders. As shown in Fig. 7 (a) and (b), the vortex flow structure for Vr=2 to 5 are not symmetric with respect to the y=0 line, leading to the displacement of cylinder 1 different from that of cylinder 2. In Fig. 7 (c) - (f), one pair of vortices are shed from each of the cylinders in one cycle of vibration and the vortex shedding flow structure is exactly symmetric with respect to the y=0 line. The symmetric vortex shedding flow results in that the vibrations of the two cylinders are in anti-phase with each other.

Fig. 7 Vorticity contours for two side-by-side elastically mounted circular cylinder in fluid

flow

f

Ay

0 1 2 30

0.1

0.2

0.3

0.4

0.5

c) Vr=06

f

Ay

0 1 2 30

0.1

0.2

0.3

0.4

0.5

d) Vr=07

X

Y

-6 -4 -2 0 2 4 6 8-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

a) Vr = 02

X

Y

-6 -4 -2 0 2 4 6-6

-5

-4

-3

-2

-1

0

1

2

3

4

b) Vr = 05

X

Y

-5 0 5-6

-4

-2

0

2

4

c) Vr = 06

X

Y

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

d) Vr = 07

X

Y

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

e) Vr = 11

X

Y

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

f) Vr = 18

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4. CONCLUSIONS

Vortex-induced vibration (VIV) of two elastically coupled side-by-side circular cylinders in steady flow at a low Reynolds number of 150 is investigated numerically. Simulations are carried out for the reduced velocities between 1 and 30, which covers the lock-in regime. The vibration of the two cylinders may lock-in with one of the two natural frequencies, depending on the reduced velocity. It is found that the lock-in regime of the two cylinder system is wider than that of the single cylinder system. The amplitude reaches its maximum when the response frequency is close to either the first mode or the second mode natural frequency. It is found that the lock-in regimes of the two natural frequencies overlap, forming a wider lock-in regime than that of a single cylinder. Similar to that of a single cylinder, the response frequency increases linearly with the reduced velocity and the response amplitude is extremely small outside the lock-in regime. 6. REFERENCES Blevins, R. D., Coughran, C. S., 2009. Experimental investigation of vortex-induced

vibration in one and two dimensions with variable mass, damping, and Reynolds number, Journal of Fluid Engineering 131, paper No. 101202.

Brika, D, & Laneville , A, 1993 Vortex-induced vibrations of a long flexible circular cylinder.J. Fluid Mech. 250, 481-508.

Feng, C. C. 1968 The measurement of vortex-induced effects in flow past a stationary and oscillating circular and D-section cylinders. Master's Thesis, University of British Columbia, Vancouver, Canada.

Fredsøe, J., Sumer, B.M., Andersen, J., Hansen, E.A., 1985. Transverse vibration of a cylinder very close to a plane wall. In: Proceedings of the Fourth International Symposium on Offshore Mechanics and Arctic EngineeinG. Dallas, ASME, vol. 1, pp. 601-609.

Gao, F.P., Yang, B., Wu, Y.X., Yan, S.M., 2006. Steady currents induced seabed scour around a vibrating pipeline. Applied Ocean Research 26, 291–298.

Govardhan, R., Williamson, C. H. K., 2000. Modes of vortex formation and frequency response for a freely-vibrating cylinder. Journal of Fluid Mechanics 420, 85-130.

Jauvtis, N., Williamson, C. H. K., 2004. The effect of two degrees of freedom on vortex-induced vibration at low mass and damping, Journal of Fluid Mechanics 509, 23-62.

Jester,W., Kallinderis, Y., 2004.Numerical study of incompressible flow about transversely oscillating cylinder pairs. Journal of Offshore Mechanics and Arctic Engineering-Transactions of the ASME 126, 310-317.

Ji, C., Xiao, Z., Wang, Y., Wang, H., 2011. Numerical investigation on vortex-induced vibration of an elastically mounted circular cylinder at low Reynolds number using the fictitious domain method. International Journal of Computational Fluid Dynamics 25 (4), 207–221.Khalak, A., Williamson, C.H.K., 1996. Dynamics of a hydroelastic cylinder with very low mass and damping. Journal of Fluids and Structures 10, 455–472.

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Khalak, A., Williamson, C. H. K., 1999. Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping, Journal of fluids and Structures 13, 813-851.Li, T., Zhang, J., Zhang, W., 2011. Nonlinear characteristics of vortex-induced vibration at low Reynolds number. Communications in Nonlinear Science and Numerical Simulation 16, 2753-2771.

Mittal, S., Kumar, Y., 1999. Finite element study of vortex-induced cross-flow and in-line oscillations of a circular cylinder at low Reynolds numbers. International Journal for Numerical Methods in Fluids 31, 1087-1120.

Mittal. S., Kumar, V., 2001. Flow-induced oscillations of two cylinders in tandem and staggered arrangements. Journal of Fluids and Structures 15(5):717.

Prasanth, T.K., Behara, S., Singh, S.P., Kumar, R., Mittal, S., 2006. Effect of blockage on vortex-induced vibrations at low Reynolds numbers. Journal of Fluids and Structures 22, 865-876.

Mittal, S., Kumar, Y., 1999. Finite element study of vortex-induced cross-flow and in-line oscillations of a circular cylinder at low Reynolds numbers. International Journal for Numerical Methods in Fluids 31, 1087-1120.

Mittal. S., Kumar, V., 2001. Flow-induced oscillations of two cylinders in tandem and staggered arrangements. Journal of Fluids and Structures 15(5):717.

Prasanth, T.K., Behara, S., Singh, S.P., Kumar, R., Mittal, S., 2006. Effect of blockage on vortex-induced vibrations at low Reynolds numbers. Journal of Fluids and Structures 22, 865-876.

Sarpakaya, T. & Isaacson, M. (1981). Mechanics of wave forces on offshore structures (first ed). Van Nostrand Reinhold.

Sarpkaya, T., 2004.A critical review of the intrinsic nature of vortex-induced vibrations. Journal of Fluids and Structures 19, 389–447.

Williamson, C.H.K., Govardhan, R., 2004. Vortex-induced vibrations.Annual Review of Fluid Mechanics 36, 413–455.

Williamson, C. H. K. & Roshko, A. 1988.Vortex formation in the wake of an oscillating cylinder. Journal of Fluids and Structures 2, 355-381. Yang, B., Gao, F.P., Jeng, D.S., Ying-Xiang Wu, Y.X., 2008. Experimental study of

vortex-induced vibrations of a pipeline near an erodible sandy seabed.Ocean Engineering 35, 301–309.

Zdravkovich, M., Pridden, D., 1977. Interference between two circular cylinders: series of unexpected discontinuities. Journal of Wind Engineering and Industrial Aerodynamics 2, 255–270.

Zhao, M., Cheng, L., 2011. Numerical simulation of two-degree-of-freedom vortex induced vibration of a circular cylinder close to a plane boundary. J. Fluids and Structures, 27, 1097-1110

Zhao, M., Cheng, L., 2012. Numerical simulation of vortex-induced vibration of four circular cylinders in a square configuration. Journal of Fluids and Structures.

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