Vortex-induced vibration (VIV) of a circular cylinder in combinedsteady and oscillatory flow

Ming Zhao a,n, Kalyani Kaja a, Yang Xiang a, Guirong Yan b

a School of Computing, Engineering and Mathematics, University of Western Sydney, Locked Bag 1797, Penrith, NSW 2751, Australiab Department of Civil Engineering, University of Texas at El Paso, TX 79968, USA

a r t i c l e i n f o

Article history:Received 18 November 2012Accepted 24 August 2013Available online 19 September 2013

Keywords:Vortex sheddingVortex-induced vibrationCircular cylinder

a b s t r a c t

Vortex-induced vibration (VIV) of a circular cylinder in the combined steady and oscillatory flow is inves-tigated numerically by solving the two-dimensional Reynolds-Averaged Navier-Stokes equations. Thefocus of the study is to investigate the effects of flow ratio, a, on the response of the cylinder. The flowratio is defined as the percentage of the steady flow velocity component in the total fluid velocity.Simulations are carried out for a constant Keulegan–Carpenter (KC) number of 10 and flow ratios rangingfrom 0 to 1 with an increment of 0.2. The reduced velocities for each flow ratio range from 2 to 25 toensure that the whole lock-in regime is covered. In the resonance regime, the frequency of the cross-flowvibration component may lock onto the oscillatory flow frequency or the natural frequency of the system,depending on the flow ratio. It locks onto twice the oscillatory flow frequency if the flow ratio ar0.2,and locks onto the natural frequency of the system if aZ0.6. It is found that the lock-in regime in thecombined steady and oscillatory flow is wider than both the one in the pure steady flow and the one inthe pure oscillatory flow. The widest lock-in regime occurs as the flow ratio is 0.4 and 0.6 and it is abouttwice as wide as that in the pure oscillatory or pure steady flow. The response at a¼0.2 is very similar tothat in the pure oscillatory flow case (a¼0), while at a¼0.8 the response is very similar to that in thepure steady flow case. At a¼0.8, the amplitude of the cylinder in the cross-flow direction reaches as highas 1.5 diameters in the “Super upper” branch because of the slowly increasing fluid velocity. It is alsofound that the vortex shedding goes through 2S, 2P and 2T modes in one period of oscillatory flow ata¼0.8 and Vr¼7, where the cross-flow amplitude is the maximum.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Vortex-induced vibration (VIV) of slender cylindrical structuresis of practical interest to many fields of engineering. For example,it influences the dynamics of offshore riser tubes bringing oil fromthe seabed to the surface. It can also cause large-amplitudevibrations of tethered structures in the ocean. The VIV of cylindershas attracted much attention from numerous researchers in thepast few decades. The forces acting on the cylinder experienceperiodic change due to the vortex shedding. The frequency of thelift force on a circular cylinder is consistent with the vortexshedding frequency, and the frequency of the drag force is doublethat of vortex shedding. Numerous experiments have shown thatin the resonance (or the lock-in), the natural frequency of anelastically mounted rigid cylinder takes control of the vortexshedding in apparent violation of Strouhal relationship. Then thefrequency of vortex shedding and the oscillation frequency of the

cylinder collapse into a single frequency, which is known as thelock-in phenomenon. Feng (1968) conducted the well-knownexperiments on the one-degree-of-freedom (1DOF) vibration of acircular cylinder in the air flow. In his study, the vibration of thecylinder was confined in the cross-flow direction and typical lock-inphenomenon was presented. In general, when a cylinder is exposedto a flow at a high mass ratio (the ratio of the mass of the cylinder tothe mass of the displaced fluid), only two amplitude responsebranches exist, i.e., initial branch and lower branch. If the cylinder isplaced in a flow at low mass ratios, the third branch (i.e., upperbranch) is observed. Khalak and Williamson (1996; 1999) and Brikaand Laneville (1993) concluded that the jump of vibration ampli-tude from the initial branch to the lower branch corresponds to amode change from 2S to 2P. Herein, 2S stands for “two singlevortices formed per vibration cycle” and 2P stands for “two pair ofvortices shedding from the cylinder per vibration cycle”. Jauvtis andWilliamson (2004) and Blevins and Coughran (2009) studied two-degree-of-freedom (2DOF) VIV of a circular cylinder and found thatthe XY-trajectory of the circular cylinder was dependent on thereduced velocity. Laneville (2006) found that the 2S and 2P vortexshedding modes were influenced by the X–Y motion of the circular

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Ocean Engineering

0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.oceaneng.2013.08.006

n Corresponding author. Tel.: þ61 2 4736 0085; fax: þ61 2 4736 0833.E-mail address: m.zhao@uws.edu.au (M. Zhao).

Ocean Engineering 73 (2013) 83–95

cylinder and the degree of this effect was related to the velocity ofthe motion. Govardhan and Williamson (2000, 2006) and Sanchiset al. (2008) carried out mode analysis of 2DOF vortex inducedvibrations. The response amplitude was found to be influenced bythe so-called Skop–Griffin parameter which is proportional tothe product of mass and damping (Griffin et al., 1975; Khalakand Williamson, 1999; Sarpkaya, 1979, 1995; Govardhan andWilliamson, 2000, 2006). Govardhan and Williamson (2006) foundthat the data collapsed very well if the effect of Reynolds numberwas taken into account as an extra parameter in a modified Griffinplot. Mittal and Kumar (1999, 2000, 2001) and Singh and Mittal(2005) simulated VIV of a circular cylinder at a low mass ratio andlow Reynolds numbers. It was found that the effects of the Reynoldsnumber on the VIVs are significant. Vibration of a circular cylinderin steady flow close to a plane boundary is also investigated due toits engineering importance (Fredsøe et al., 1985; Gao et al., 2006;Yang et al., 2006, 2008; Zhao and Cheng, 2011). It was found thatthe plane boundary had significant effects on the vibration.

When wave-induced hydrodynamic loads on subsea structuressuch as pipelines or risers are examined, wave motions arecommonly modelled as oscillatory flows. Studies have shown thathydrodynamic forces and flow characteristics around a circularcylinder in a sinusoidal oscillatory flow are dependent on theKeulegan–Carpenter (KC) number, which is defined as KC¼UmT/D,where Um is the amplitude of the oscillatory flow velocity, T is theoscillatory flow period and D is the diameter of the cylinder. WhenKC is sufficiently small, the boundary-layer around the cylindersurface is laminar and two-dimensional. As the KC numberincreases, the flow in the laminar boundary layer becomesunstable and the flow becomes three-dimensional. Vortex shed-ding occurs at large KC numbers, resulting in hydrodynamic liftforces in the transverse direction of the flow. Govardhan andWilliamson (2006) experimentally visualized flow characteristicsaround a circular cylinder in an oscillatory flow with KC numbersranging from 4 to 30. It was observed that the number of vortexpairs that were shed from the cylinder in a flow period increasedwith increasing KC numbers. The oscillatory flow around a circularcylinder was classified into the pairing of attached vortices,transverse street (7oKCo13), single pair (13oKC o15), doublepair (15oKCo24), three pairs (24oKCo32) and four pairs(32oKCo40) vortex shedding regimes, based on the number ofvortex pairs shed during each half of a flow period. Govardhan andWilliamson (2006) also found the relationship between vortexmotions and time-dependent lift-force variations in each vortexshedding regime. Kozakiewicz et al. (1997) and Sumer and Fredsøe(1988) carried out experiments of 1DOF vibration of an elasticallymounted cylinder exposed to an oscillatory flow in the cross-flowdirection. The experimental results show that the response patternat a constant KC number varies with the reduced velocity. One ofthe typical characteristics of the response in the oscillatory flow isthat the vibration frequency is multiple of the oscillatory flowfrequency. Kozakiewicz et al. (1992) and Sumer et al. (1994)studied the span wise correlation of a vibrating cylinder inoscillatory flow and found that the correlation along the cylinderincreased with amplitudes and the maximum correlation isattained when the vibrations are in the lock-in regime. Somenumerical studies are also carried out to study the VIV of a circularcylinder in an oscillatory flow. Guilmineau and Queutey (2002)simulated vortex shedding flow in the wake of a circular cylinderundergoing forced cross-flow vibrations. Anagnostopoulos andIliadis (1998) carried out numerical study on the 1DOF VIV of acircular cylinder in the stream wise direction and found that theresponse of the cylinder was amplified significantly if the oscilla-tory flow frequency is close to the natural frequency of thecylinder. Zhao et al. (2012) simulated 1DOF VIV of a cylinder inoscillatory flows at KC¼10 and 20 and found that the response of

the cylinder had more than one frequency components as thereduced velocity exceeded 8.

The offshore structures have to withstand the impacts ofcombined currents and waves, which can be modelled by com-bined steady and oscillatory flows in the numerical studies. In thispaper, VIV of a circular cylinder in the combined steady andoscillatory flow at a low mass ratio of 2.5 is investigated numeri-cally. The fluid flow is simulated by the two-dimensional Rey-nolds-averaged Navier–Stokes (RANS) equations and the shearstress transport (SST) k–ω turbulence model (Menter, 1994) is usedto simulate the turbulence. The effects of flow ratio, a, on theresponse of the cylinder are studied. The flow ratio is defined asthe percentage of the steady flow component in the total fluidvelocity. Simulations are carried out for a constant Keulegan–Carpenter (KC) number of 10 and flow ratios ranging from 0 to1 with an increment of 0.2. The reduced velocities for each flowratios range from 2 to 25 to ensure that the lock-in regime is fullycovered.

2. Numerical method

Vortex-induced vibration of a circular cylinder in the combinedsteady and oscillatory flow as shown in Fig. 1(a) is considered. Thefluid velocity u(t) in the combined flow is expressed as

uðtÞ ¼UcþUm cos ð2πt=TÞ ð1Þ

where t is time, Uc is the steady flow velocity, Um and T are theamplitude and period of the oscillatory flow velocity, respectively.The governing equations for simulating the turbulent flow arethe unsteady two-dimensional incompressible Reynolds-AveragedNavier–Stokes (RANS) equations. In this study, the Arbitrary

Fig. 1. (a) Computational domain and (b) the computational mesh close to thecylinder surface.

M. Zhao et al. / Ocean Engineering 73 (2013) 83–9584

Lagrangian Eulerian (ALE) scheme is applied to deal with the movingboundaries of the cylinder. The velocity, length, time and pressure arenon-dimensionlized by (Zhao et al., 2013)

ui ¼~ui

f nD; xi ¼

~xiD; t ¼ f n ~t ; p¼ ~p

ρf 2nD2; ð2Þ

where x1¼x and x2¼y are the Cartesian coordinates in the in-lineand transverse directions of the flow respectively, D is the diameterof the cylinder, ui is the fluid velocity component in the xi direction, pis the pressure, ρ is the fluid density, ν is the kinematic viscosity andfn is the structural natural frequency of the cylinder system. The tildesover the symbols in Eq. (2) denote the dimensional variables. Thenondimensional incompressible RANS equations can be expressed by

∂ui

∂xi¼ 0 ð3Þ

∂ui

∂tþðuj�ujÞ

∂ui

∂xj¼�∂p

∂xiþVr

Re∂2ui

∂x2iþ ∂∂xj

ð�u′iu

′j Þ; ð4Þ

where ui is the velocity of the mesh movement, Re is the Reynoldsnumber defined by Re¼(UcþUm)D/ν and Vr is the reduced velocitydefined by Vr¼(UcþUm)/(fnD). The normalization for the RANSequations results in the nondimensional combined free-streamvelocity (non-dimensional UcþUm) equaling to the reduced velocityand the nondimensional vibration frequency of f ¼ f ′=f n with f ′being the dimensional vibration frequency. The Reynolds stresstensor u′iu′j is computed by

�u′iu′j ¼ νtð∂ui=∂xjþ∂uj=∂xiÞþ23kδij; ð5Þ

where νt is the turbulent viscosity and k is the turbulent energy. Theshear stress transport (SST) k–ω turbulence model (Menter, 1994) isused for modelling the turbulence. It was found that the SST k–ωmodel gives a good prediction of the adverse pressure gradient flows.The RANS equations are discretized by the Petrov–Galerkin FiniteElement Method (FEM) developed by the first author of this paper(Zhao et al., 2007). In the Petrov–Galerkin FEM, the standard Galerkinweighting functions are modified by adding a streamline upwindperturbation, which acts only in the flow direction (Brooks andHughes, 1982). A rectangular computational domain of a nondimen-sional width of 60 and a height of 40 is discretized into finite elementmodels. Fig. 1(b) shows the computational mesh in the vicinity of thecylinder. The surfaces of the cylinder are assumed to be smooth,where no-slip boundary condition is imposed. Specifically, the fluidvelocity along the cylinder surface is the same as the vibrating speedof the cylinder. On the cylinder surface, the turbulent energy k is zeroand the specific dissipation rate ω is given at the nodal points next tothe wall surface as ω¼ 6Re=Δ2

1, where Δ1 is the distance from thewall. The inlet velocity boundary conditions are set as u¼ Vr , v¼ 0.The nondimensional turbulence quantities is k¼ 0:001V2

r and ω¼ 1(Guilmineau and Queutey, 2004) at inlet boundary. At the outflowboundary, the gradients of fluid velocity and turbulent quantities inthe direction normal to the boundary are set to be zero. Pressure atthe outflow boundary is given a reference value of zero. The two-degree-of-freedom equation of the motion for the displacements ofthe cylinder system is given by

∂2Xi

∂t2þ4πζ

∂Xi

∂tþ4π2Xi ¼

2π

V2r

mnCFi; ð5Þ

0.0

0.5

1.0

1.5

2.0

2.5

2 4 6 8 10 12 14

Present

Increasing velocity, Jauvtis and Williamson, 2004Constant velocity, Zhao et al., 2011

Increasing velocity, Zhao et al., 2011

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14

Constant velocity, PresentIncreasing velocity, Jauvtis and Williamson, 2004

Fig. 2. Comparison between the numerical results of 2DOF VIV in pure steady flowwith the experimental data. (a) Cross-flow amplitude and (b) Cross-flow frequency.

0.0

0.5

1.0

1.5

0 2 4 6 8 10 12 14

0.0

0.5

1.0

1.5

2.0

0 2 4 6 8 10 12 14

Fig. 3. Comparison between the numerical results of 2DOF VIV in pure oscillatoryflow with the experimental data KC¼10. (a) Cross-flow amplitude and (b) Cross-flow frequency.

M. Zhao et al. / Ocean Engineering 73 (2013) 83–95 85

where X1 ¼ X and X2 ¼ Y are the cylinder displacements in thex-direction and the y-direction respectively. The mass ratio mn, thestructural damping ratio ζ and the force coefficient CFi are defined as

mn ¼ mmd

; ζ¼ c

2ffiffiffiffiffiffiffiffiKm

p ; CFi ¼Fi

ρDU2=2ð6Þ

where m is the mass of the cylinder, md the displaced fluid mass,c the structural damping constant, K is the structural stiffness and Fithe hydrodynamic force on the cylinder in the xi direction. In thisstudy the structural stiffness in the x-direction is the same as that inthe y-direction, and the structural damping ratios in both the x- andy-directions are set to be zero. The temporal integration of Eq. (6) isperformed numerically by the fourth-order Runge–Kutta algorithm.After each computational time step, the boundary of the computa-tional domain changes because of the displacement of the cylinders.The positions of finite element nodes are moved accordingly bysolving the modified Laplace equation (Zhao and Cheng, 2011)

∇ γ∇Sið Þ ¼ 0; ð7Þ

where Si represents the displacement of the nodal points in the xidirection, and γ is a parameter that controls the mesh deformation.In order to avoid excessive deformation of the near-wall elements, theparameter γ in an finite element is set to be γ ¼ 1=A, with A being thearea of the element. The displacement of the mesh nodes is the sameas the displacement of the cylinder on the cylinder surface and zero onother boundaries. By giving the displacements at all the boundaries,Eq. (7) is solved by a Galerkin FEM. Initially, the velocity and thepressure are zero in the whole computational domain and thecylinder's displacement and velocity are zero in all the simulations.

3. Numerical results

2DOF VIV of a circular cylinder in the combined steady andoscillatory flow at a constant KC number of 10 and a constant mass

-1-0.5

00.5

1

0 5 10 15 20 25 30 35 40

X, Y

t/T

X Y

-3-2-10123

0 5 10 15 20 25 30 35 40 45

X Y

-1

0

1

X, Y

X Y

-3-2-10123

0 4 8 12 16 20 24 28 32 36

X Y

-1

0

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

X, Y

X Y

-2-1012 X Y

-1

0

1

0 2 4 6 8 10 12 14 16 18

X, Y

X Y

-2-10123 X Y

-2-1012

0 1 2 3 4 5 6 7 8 9

X, Y

X Y

-1

0

1

2 X Y

-2-1012

20 25 30 35 40 45 50 55 60 65 70 75 80

X, Y

t

X Y

-1

0

1

2

0 5 10 15 20 25 30 35t

X Y

X, Y

X, Y

X, Y

X, Y

X, Y

X, Y

0 5 10 15 20 25 30 35 40

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

0 2 4 6 8 10 12 14 16 18

0 1 2 3 4 5 6 7 8 9

t/T

t/T

t/T

t/T t/T

t/T

t/T

t/T

t/T

Fig. 4. Time histories of the displacements of the cylinder. (a) a=0, Vr=5, (b) a=0, Vr=8, (c) a=0.2, Vr=6, (d) a=0.2, Vr=10, (e) a=0.4, Vr=6, (f) a=0.4, Vr=10, (g) a=0.6, Vr=5, (h)a=0.6, Vr=16, i) a=0.8, Vr=7, (j) a=0.8, Vr=14, (k) a=1, Vr=5, (l) a=1 and Vr=12.

M. Zhao et al. / Ocean Engineering 73 (2013) 83–9586

ratio of 2.5 is simulated numerically. The study is focused on theeffects of the flow ratio on the response of the cylinder. The flowratio a is defined as a¼Uc/(UcþUm). The Reynolds number is fixedto be 5000 in all the simulations. Simulations are carried out forthe reduced velocities ranging from 2 to 25 with an interval of1 and the flow ratios ranging from 0 to 1 with an interval of 0.2.The damping factor of the cylinder is equal to zero in thenumerical simulation. Fig. 1 shows the nondimensional computa-tional domain and the computational mesh near the cylinder. The60 long and 40 wide non-dimensional computational domain isdivided into 20946 quadrilateral linear finite elements. Refinedelements are used near the cylinder surface in order to model theboundary layer flow. The circumference of the cylinder is dividedinto 96 nodes and the nondimensional minimum mesh size in theradial direction is 0.0008. The computational time step Δt ischosen based on the criteria VrΔt/D¼0.001. The mesh dependencyhas been carried out and it has been found that further increase inthe mesh density makes little change on the numerical results.

3.1. Comparison with the experimental data

The numerical results of the VIV of a circular cylinder in thepure steady flow are compared with the experimental data byJauvtis and Williamson (2004) in Fig. 2. Two reduced velocities aredefined in this study. The reduced velocity based on the structuralnatural frequency Vr is defined as Vr ¼ ðUcþUmÞ=f nD, and thereduced velocity based on the natural frequency in water Vrw isdefined as Vrw ¼ ðUcþUmÞ=f nwD. In order to compare the numer-ical results with the experimental data straightforwardly, theresponse of the cylinder is plotted against the reduced velocitythat is based on the natural frequency in water. The relationshipbetween the natural frequency in water (fnw) and the structuralnatural frequency is f nw ¼ f n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimn=ðCAþmnÞ

p, where CA¼1 is the

added-mass coefficient (Sumer and Fredsøe, 1997). The vibrationamplitudes are defined as Ax ¼ ðXmax�XminÞ=2D andAy ¼ ðYmax�YminÞ=2D, where Xmax and Xmin are the maximum

and the minimum displacements in the x-direction in 10 periodsof oscillatory flow (10 periods of vibration in the pure steadycurrent case, i.e. a¼1), and Ymax and Ymin are the correspondingvalues in the y-direction. In the experiments by Jauvtis andWilliamson (2004), the mass ratio was 2.6 and the mass-damping parameter is (mnþCA)ζ¼0.013. By using the increasingvelocity condition a “super upper” branch is observed where theamplitudes in the cross-flow direction reached 1.5 cylinder dia-meters (Jauvtis and Williamson, 2004). The “super upper” branchcan only be achieved by increasing the velocity gradually in theexperiments. The numerical model used in this study is exactly thesame as that used in Zhao et al. (2011). Zhao et al. (2011)reproduced the “super upper” branch numerically by increasingthe velocity slowly. If the velocity is given a constant value at thebeginning of the numerical simulation, the “super upper” branchcannot be achieved. The present numerical results of the cross-flow amplitude agree well with the experimental data, except inthe “super upper” branch. Zhao et al. (2011) demonstrated that the“super upper” branch cannot be predicted numerically if theconstant velocity condition is applied. The slight differencebetween the numerical results in this study and that in Zhaoet al. (2011) is due to the difference in the Reynolds number. In thisstudy, the Reynolds number is fixed to be 5000, while theReynolds number in Zhao et al. (2011) ranged from 1000 to14000. The variation of the calculated vibration frequencies isconsistent with the experimental data except in the “super upper”branch.

Fig. 3 shows the comparison of the calculated response of thecircular cylinder in the pure oscillatory flow with the experimentaldata, where f w ¼ 1=T is the frequency of the oscillatory flow. Sinceno experimental data of 2DOF VIV in the oscillatory flow areavailable, the experimental data of 1DOF VIV in the cross-flowdirection are shown in Fig. 3 for comparison. In both 1DOF and2DOF cases, the maximum vibration amplitude occurs at thereduced velocity of about Vr¼6. Same as what was found in theexperiments by Sumer and Fredsøe (1988) and Kozakiewicz et al.

Fig. 5. Trajectories of the vibration.

M. Zhao et al. / Ocean Engineering 73 (2013) 83–95 87

(1997), the vibration frequency of the cylinder is twice theoscillatory flow frequency as 4rVrr10, where the vibrationamplitude in the cross-flow direction is significantly large. In this

study the term “lock-in” also applies to the case where the cross-flow vibration frequency is the same as or twice the oscillatoryflow frequency. The cross-flow amplitude as VrZ7 in the 2DOF

0

0.2

0.4

0.6

0 1 2 3f

0

0.2

0.4

0.6

0 1 2 3f

0

0.2

0.4

0.6

0 1 2 3f

0

0.2

0.4

0.6

0 1 2 3f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0.8

1

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0.8

1

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0.8

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0.8

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

0

0.2

0.4

0.6

0.8

0 1 2 3

f

0

0.2

0.4

0.6

0 1 2 3

f

Fig. 6. Spectra of the displacement in the cross-flow direction. (a) a=0, Vr=3, (a) a=0, Vr=3, (b) a=0, Vr=4, (c) a=0, Vr=5, (d) a=0, Vr=10 (e) a=0.2, Vr=5, (f) a=0.2, Vr=7,(g) a=0.2, Vr=10, (h) a=0.2, Vr=11, (i) a=0.4, Vr=8, (j) a=0.4, Vr=12, (k) a=0.4, Vr=14, (l) a=0.4, Vr=18, (m) a=0.6, Vr=5, (n) a=0.6, Vr=10, (o) a=0.6, Vr=15, (p) a=0.6, Vr=18,(q) a=0.8, Vr=5, (r) a=0.8, Vr=8, (s) a=0.8, Vr=10, (t) a=0.8, Vr=12, (u) a=1, Vr=4, (v) a=1, Vr=6, (w) a=1, Vr=8 and (x) a=1, Vr=10.

M. Zhao et al. / Ocean Engineering 73 (2013) 83–9588

VIV is much larger than that in 1DOF VIV. Outside the lock-inregime (VrZ13), the vibration amplitude is close to zero and thevibration frequency is the same as the oscillatory flow frequency.

3.2. VIV of a circular cylinder in the combined steady and oscillatoryflow

Fig. 4 shows the time histories of the displacements of thecircular cylinder at some typical reduced velocities in the lock-inregime. For each flow ratio, time histories at two reduced velo-cities are presented. When it is in the pure steady flow (a¼1) or inthe pure oscillatory flow (a¼0), the circular cylinder alwaysvibrates periodically and predominantly at one frequency in thecross-flow direction. However, in the combined steady and oscil-latory flow (0oao1), the vibration of the cylinder may becomeirregular especially in the cross-flow direction (Fig. 4 (f)). Beatingis the typical phenomenon found at a¼0.6 and 0.8. In oneoscillatory flow period, the vibration amplitude in the cross-flowdirection increases and decreases periodically at the beatingfrequency, which happens to be the same as the frequency ofthe oscillatory flow. It is believed that the beating occurs becausethe fluid velocity increases and decreases periodically. If the KCnumber is kept constant, the oscillatory flow velocity componentis small and the period is large at large flow ratios. The total

velocity is always positive because the steady current componentdominates. The slow variation of the flow velocity with a longperiod allows the change in vibration mode of the circular cylinder(due to the change of the velocity) within one period of oscillatoryflow as discussed later on. The periodical change in vibration modeleads to the beating in the response.

Fig. 5 shows the XY-Trajectories of the VIV at typical reducedvelocities in the lock-in regimes. The “V” shaped trajectory at a¼0and Vr¼4 and the “1” shaped trajectory at a¼0 and Vr¼5 in thepure oscillatory flow are the same as what have been identified byZhao et al. (2011). The vibration trajectories at a¼0.2 are similar tothose at a¼0. However, each “1” trajectory at a¼0.2 is asym-metric. The vibration trajectories at a¼0.8 and Vrr10 are verysimilar to those at a¼1 (the pure steady flow case), where thecylinder mainly vibrates in the cross-flow direction. The vibrationamplitude in the in-line direction increases with Vr and is greaterthan that in the cross-flow direction as VrZ14 for all the flowratios. At a¼0.8 and Vr¼16, the cylinder mainly vibrates in the in-line direction and the vibration in the cross-flow direction has avery small amplitude and higher frequency than the oscillatoryflow frequency. It can be seen in Fig. 5 that, for all the cases wherea is less than 1, the vibration in the in-line direction is moresignificant that in the cross-flow direction at large reducedvelocities.

0.0

0.5

1.0

1.5

2.0

0 5 10 15 20 250.0

0.5

1.0

1.5

2.0

0 5 10 15 20 25

0.0

0.5

1.0

1.5

2.0

0 5 10 15 20 250.0

0.5

1.0

1.5

2.0

0 5 10 15 20 25

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20 250.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20 25

Fig. 7. Variation of the vibration frequency in the cross-flow direction with the reduced velocity. (a) a=0, (b) a=0.2, (c) a=0.4, (d) a=0.6, (e) a=0.8 and (f) a=1.

M. Zhao et al. / Ocean Engineering 73 (2013) 83–95 89

The vibration frequency in the in-line direction is dominated bythe oscillatory flow frequency as the flow ratio ar0.8. However,the vibration frequency in the cross-flow direction is not the same.Fast Fourier Transform (FFT) is applied to analyze the frequency ofthe vibration displacement in the cross-flow direction. The spectraare obtained by performing the FFT on at least 10 oscillatory flowperiods of the equilibrium numerical results in the combinedoscillatory and steady flow cases (αr0.8) Fig. 6 shows theamplitude spectra of the cross-flow displacement for some typicalreduced velocities. Four spectra are shown for each flow ratio.It can be seen that single distinct peak frequency dominates ineach spectrum for a¼0, 0.8 and 1. The frequency corresponding tothe highest peak in a spectrum is defined as the primary peakfrequency and that corresponding to the secondary peak is definedas the secondary peak frequency. It can be seen that the primarypeak frequency is close to 1 in most of the spectra in Fig. 6.As a¼0.4 and 0.6, where the oscillatory flow velocity is close to thesteady flow velocity, the spectra are broad-banded at somereduced velocities (Fig. 6 (i), (k) and (o)), indicating the irregularityof the vibration.

Fig. 7 presents the variation of the vibration frequency with thereduced velocity at different flow ratios. It should be noted that,based on the nondimensionalization method shown in Eq. (2), thenondimensional frequency fy in this study is the ratio ofthe vibration frequency to the structural natural frequency. Theprimary peak frequency (the peak frequency with the highestamplitude) is shown in Fig. 7 if a spectrum has more than onepeaks. Fig. 8 shows the variation of the vibration amplitude withthe reduced velocity at different flow ratios. The amplitude in the

lock-in regime of Vr is generally between 0.6 and 0.7 in the puresteady flow case (a¼1). It can be seen in Fig. 7 (a) that in the pureoscillatory flow case, the cylinder vibrates at a frequency that iseither equal to or twice the oscillatory flow frequency, dependingon the reduced velocity. Kozakiewicz et al. (1997), Sumer andFredsøe (1988) and Zhao et al. (2012) found that a circular cylinderin a purely oscillatory flow vibrates in the cross-flow direction atfrequencies that are multiples of the oscillatory flow frequency.Zhao et al. (2012) defined the vibration mode according to thefrequency of the vibration. If a cylinder vibrates predominantly ina frequency the same as the oscillatory flow frequency, thevibration is in the single-frequency mode. If the vibration fre-quency of the cylinder is twice the oscillatory flow frequency, thevibration is in the double-frequency mode and the rest may bededuced by analogy. In this study, it was found that the vibrationof the cylinder changes from the double-frequency mode to thesingle-frequency mode at Vr¼10 in the pure oscillatory flow case.The vibration amplitude has almost been zero at VrZ12 and a¼1.

At a¼0.2, the flow changes from the double-frequency mode tothe single frequency mode at Vr¼11 and the vibration amplitudein the cross-flow direction is still significant as Vr is between 11and 14. At a¼0.4, the frequency of the vibration in the cross-flowdirection is multiple of the oscillatory flow frequency only occa-sionally. It is in the double-frequency mode at 9rVrr11 and inthe single-frequency mode at Vr¼12, 13, 16–18 and 24, 25.It appears that with the increase in Vr, the vibration frequency inthe cross-flow direction locks onto the natural frequency (f isclose to 1) if it is neither the single-frequency nor the double-frequency mode. At a¼0.6, the characteristic of the vibration that

0.0

0.5

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2.0

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plitu

de

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plitu

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plitu

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Am

plitu

de

0 5 10 15 20 25

0 5 10 15 20 25 0 5 10 15 20 25

0 5 10 15 20 250 5 10 15 20 25

Vr

Vr Vr

Vr

Vr

Fig. 8. Variation of the vibration amplitudes with the reduced velocity. (a) a=0, (b) a=0.2, (c) a=0.4, (d) a=0.6, (e) a=0.8 and (f) a=1.

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the cross-flow frequency is either the single- or double-frequencyhas totally vanished. Instead, the vibration frequency is close to thenatural frequency in a regime of Vr which is much wider than boththat in the pure oscillatory flow case and that in the oscillatoryflow case.

Although the regimes of Vr for lock-in are different, thevariations of amplitudes with the reduced velocity at a¼0 to0.8 are very similar to each other. As 0rar0.8, the vibrationamplitude in the cross-flow direction is comparable with that inthe in-line direction as the reduced velocity is smaller than acritical value. Once the reduced velocity exceeds this critical value,the cross-flow amplitude is smaller than that in the in-linedirection and the difference between the two increases with theincrease in Vr. The variations of the cross-flow amplitude and thecross-flow frequency with reduced velocities at a¼0.8 are verysimilar to those at a¼1. The cross-flow frequency is close to thenatural frequency in the lock-regime and increases with theincrease in Vr outside the lock-in regime. The maximum amplitudeat a¼0.8 in the lock-in regime is about 1.5 which is much largerthan that in the pure steady flow case of a¼1. Jauvtis andWilliamson (2004) reported that the cross-flow amplitude in the“super upper” branch can reach as high as 1.5 cylinder diameters ifthe velocity is increased gradually. The “super upper” branch isdefined as the response of the reduced velocity regime where thecross-flow amplitude increases to as high as 1.5 times cylinder

diameters. Similar to what was observed by Jauvtis andWilliamson (2004), the vibration frequency in the cross-flowdirection is slightly less than 1 in the “super upper” branch, wherethe cross-flow amplitude increases with the increase in Vr andslightly less than 1 in the “lower branch”, where the cross-flowamplitude stops increasing with the increase in Vr (Fig. 3(e) and(f)). The “super upper branch” is only found at a¼0.8 and not ata¼1, indicating that introducing a small variation (20% of the totalvelocity) to the fluid velocity increases the amplitude of thevibration significantly. It can be seen in Fig. 8(e) that the in-lineamplitude also reaches its maximum value at the upper boundaryof the “super upper” branch. In the “lower branch”, the in-lineamplitude increases with Vr. At the high flow ratio of a¼0.8, thefluid velocity increases gradually for a relatively long period oftime because the small oscillatory flow component and the longoscillatory flow period, resulting in the “super upper” branch withan cross-flow amplitude of about 1.5. The variation of the vibrationfrequency in the cross-flow direction with the reduced velocity ata¼0.8 is similar to that in the steady flow case, i.e., the cross-flowfrequency is slightly lower than 1 in the “super upper” branch andslightly higher than 1 in the “lower branch”. The response of thevibration in the in-line direction at a¼0.8 is different from that ata¼1. The in-line amplitude at a¼0.8 increases with Vr in the“super upper” branch and suddenly decreases to zero at the higherboundary of the “super upper” branch. In the lower branch the

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40.5 41 41.5 42 42.5 43 43.5 44

X, Yu

t/T

u X Y

a

d

efc

b o

Fig. 9. Vorticity contours within one oscillatory flow period for a¼0 and Vr¼6. (a) t/T=41.59, (b) t/T=41.71, (c) t/T=41.92, (d) t/T=42.19 (e) t/T=42.46 and (f) t/T=42.60.

M. Zhao et al. / Ocean Engineering 73 (2013) 83–95 91

in-line amplitude at a¼0.8 increases with Vr again. However, thein-line amplitude at a¼1 is always negligibly small compared withthe cross-flow amplitude except Vr¼2, which is outside the lock-inregime.

Fig. 9 shows the vorticity contours in six instants within oneoscillatory flow period for a¼0 and Vr¼6. The instants for thevorticity contours are indicated in the time histories of the cross-flow displacement shown at the bottom of Fig. 9. When astationary circular cylinder is in the pure oscillatory flow, one pairof vortices are shed from the cylinder if the KC number is 10 and

the two vortices are shed from the same side of the cylinder,forming the so-called transverse vortex street (Williamson 1985).In Fig. 9, the number of the vortices that are shed from the cylinderis also one pair (the negative vortex in Fig. 9(b) and the positivevortex in Fig. 9(d)) and both vortices are shed from the bottom ofthe cylinder. All other vortices do not grow strong enough beforethe flow reverses. After the flow reverses, the vortices that havenot shed from the cylinder dissipate. Only one vortex has thechance to grow and shed from the cylinder in half of theoscillatory flow period at KC¼10.

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27 27.5 28 28.5 29 29.5 30

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t/T

u X Y

ad

e

fc

b

gh

o

Fig. 10. Vorticity contours within one oscillatory flow period for a¼0.4 and Vr¼9. (a) t/T=27.50, (b) t/T=27.67, (c) t/T=27.81, (d) t/T=27.93 (e) t/T=28.06, (f) t/T=28.24.(g) t/T=28.37 and (h) t/T=28.47.

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Fig. 11. Vorticity contours within one oscillatory flow period for a¼0.8 and Vr¼7. (a) t/T=8.54, (b) t/T=8.64, (c) t/T=8.72, (d) t/T=8.78 (e) t/T=8.85, (f) t/T=8.93. (g) t/T=8.96,(h) t/T=9.03, (i) t/T=9.11, (j) t/T=9.18, (k) t/T=9.28, (l) t/T=9.34, (m) t/T=9.38, (n) t/T=9.45 and (o) t/T=9.52.

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Fig. 10 shows the vorticity contours within one oscillatory flowperiod for a¼0.4 and Vr¼9. Within one oscillatory flow period, themaximum fluid velocity in the positive x-direction is larger thanthat in the negative x-direction. The duration when the flowvelocity is positive is also longer than that when the flow isnegative. Like the case of a¼0, the number of the vortices that areshed from the cylinder is also two. However, both vortices are shedfrom the cylinder when the fluid flow is in the positive x-direction(Fig. 10(e) and (f)).

Fig. 11 shows the vorticity contours within one oscillatory flowperiod at a¼0.8 and Vr¼7, where the cross-flow amplitude is themaximum among other reduced velocities. An interesting phe-nomenon at the flow ratio of a¼0.8 is beating, as shown in Fig. 4(i) and (j). It is found in Fig. 11 that the vortex shedding goesthrough different modes in an oscillatory flow period. In Fig. 11(b) to (e), where the fluid velocity is increasing to its maximumvalue gradually, two vortices are shed from the cylinder in onecross-flow vibration cycle. Between Fig. 11(g) and (i), where thefluid velocity starts decreasing gradually to its minimum value,the vortex shedding is found to be a triplet pair (2T) regime. Whenthe cylinder is moving downwards, three vortices are shed from it(marked by A1 to A3 in Fig. 11(g)) and another three vortices areshed from it when the cylinder is moving upwards (marked by B1to B3 in Fig. 11(h)). Jauvtis and Williamson (2004) reported thatthe 2T regime can be achieved by increasing the fluid velocitygradually. The 2T regime sustains for two cross-flow vibrationcycles until Fig. 11(i). The triple pair vortex shedding mode in twocross-flow vibration cycles is followed by a cycle of double pairmode (two vortices are marked by C1 and C2 in Fig. 11(m) andanother two are marked by D1 and D2 in Fig. 11(n)). After Fig. 11(n), the flow reverses and a new cycle of single pair, triple pair anddouble pair repeats.

4. Conclusions

VIV of a circular cylinder in combined steady and oscillatoryflows is investigated by a numerical method. In this study,extensive simulation is carried out for KC¼10 and the flow ratiosranging from 0 to 1 with an interval of 0.2. The range of thereduced velocity Vr is between 2 and 25, which covers the fulllock-in regime for all the flow ratios. It is found that the flow ratiohas significant effects on the response of the cylinder. The mostimportant finding in this study is that the combination of thesteady flow and the oscillatory flow widens the lock-in regime.The widest lock-in regimes, which occur at a¼0.4 and 0.6, areabout twice as wide as that in the pure steady or pure oscillatoryflow case. At KC¼10, the response of the cylinder in the cross-flowdirection locks onto either twice the oscillatory flow frequency orthe natural frequency of the structure, depending on the flowratio. It locks onto twice the oscillatory flow frequency if the flowratio ar0.2 in the resonance regime, and locks onto the naturalfrequency of the system as aZ0.6. The response at a¼0.2 is verysimilar to that in the pure oscillatory flow case, while that ata¼0.8 is very similar to the pure steady flow case. At a¼0.4, thevibration frequency in the cross-flow direction generally locksonto the natural frequency of the cylinder and occasionally locksonto the oscillatory flow frequency or twice the oscillatory flowfrequency. At a¼0.6, the vibration frequency in the cross-flowdirection predominantly locks onto the natural frequency of thecylinder.

Among all the flow ratios, the highest cross-flow amplitudeoccurs at a¼0.8, which is about 1.5 times cylinder diameters.At a¼0.8, the fluid velocity increases and decreases slowly withinone oscillatory flow period. As demonstrated by previous studies,the gradually increasing velocity leads to the so-called “super

upper” branch that has maximum cross-flow amplitude of 1.5 timesthe cylinder diameters (Jauvtis and Williamson, 2004). At a¼0.8 andVr¼7, where the cross-flow amplitude is the maximum, the cylindervibrates a number of cycles in the cross-flow direction and the vortexshedding goes through 2S, 2P and 2T modes within a single oscillatoryflow period, resulting in the beating phenomenon in the response.

References

Anagnostopoulos, P., Iliadis, G., 1998. Numerical study of the flow pattern and thein-line response of a flexible cylinder in an oscillating stream. Journal of FluidsStructures 12, 225–258.

Blevins, R.D., Coughran, C.S., 2009. Experimental investigation of vortex-inducedvibration in one and two dimensions with variable mass, damping, andReynolds number. Journal of Fluids Engineering131, 101202.

Brika, D, Laneville, A, 1993. Vortex-induced vibrations of a long flexible circularcylinder. Journal of Fluid Mechanics 250, 481–508.

Brooks, A.N., Hughes, T.J.R., 1982. Streamline upwind/Petrov–Galerkin formulationsfor convection dominated flows with particular emphasis on the incompres-sible Navier–Stokes equations. Computer Methods in Applied Mechanics andEngineering 32, 199–259.

Feng, C.C., 1968. The measurement of vortex-induced effects in flow past astationary and oscillating circular and D-section cylinders. University of BritishColumbia, Vancouver, Canada. (Master's thesis).

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

Gao, F.P., Yang, B., Wu, Y.X., Yan, S.M., 2006. Steady current induced seabed scouraround a vibrating pipeline. Applied Ocean Research 28 (5), 291–298.

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

Govardhan, R., Williamson, C.H.K., 2006. Defining the ‘modified Griffin plot invortex-induced vibration: revealing the effect of Reynolds number usingcontrolled damping. Journal of Fluid Mechanics 561, 147–180.

Griffin, O. M., Skop, R. A. & Ramberg, S. E. 1975 The resonant vortex-excitedvibrations of structures and cable systems. In Proceedings of the 7th OffshoreTechnology Conference, Houston, TX, OTC Paper 2319.Vortex-excited vibrationof a circular cylinder in planar oscillating flow.

Guilmineau, E., Queutey, P., 2002. A numerical simulation of vortex shedding froman oscillating circular cylinder. Journal of Fluids and Structures 16 (6), 773–794.

Guilmineau, E., Queutey, P., 2004. Numerical simulation of vortex-induced vibrationof a circular cylinder with low mass-damping in a turbulent flow. Journal ofFluids and Structures 19, 449–466.

Jauvtis, N., Williamson, C.H.K., 2004. The effect of two degrees of freedom onvortex-induced vibration at low mass and damping. Journal of Fluid Mechanics509, 23–62.

Khalak, A., Williamson, C.H.K., 1996. Dynamics of a hydroelastic cylinder with verylow mass and damping. Journal of Fluids and Structures 10, 455–472.

Khalak, A., Williamson, C.H.K., 1999. Motions, forces and mode transitions invortex-induced vibrations at low mass-damping. Journal of fluids and Struc-tures 13, 813–851.

Kozakiewicz, A., Sumer, B.M., Fredsøe, J., 1992. Spanwise correlation on a vibratingcylinder near a wall in oscillatory flows. Journal of Fluids and Structures 6,371–392.

Kozakiewicz, A., Sumer, B.M., Fredsøe, J., 1997. Vortex regimes around a freelyvibrating cylinder in oscillatory flow. International Journal of Offshore andPolar Engineering 7 (2), 102–104.

Laneville, A., 2006. On vortex-induced vibrations of cylinders describing X-Ytrajectories. Journal of Fluids and Structures 22, 773–782.

Menter, F.R., 1994. 2-equation eddy-viscosity turbulence models for engineeringapplications. AIAA Journal 32 (8), 1598–1605.

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

Mittal, S., Kumar, V., 2000. Flow-induced oscillations of two cylinders in tande-mand staggered arrangement. Journal of Fluids and Structures 15, 717–736.

Mittal, S., Kumar, V., 2001. Flow induced vibrations of a light circular cylinder atReynolds numbers 103 to 104. Journal of Sound and Vibration 245, 923–946.

Sanchis, A., Sælevik, G., Grue, J., 2008. Two-degree-of-freedom vortex-inducedvibrations of a spring-mounted rigid cylinder with low mass ratio. Journal ofFluids and Structures 24, 907–919.

Sarpkaya, T., 1979. Vortex-Induced Oscillations. A Selective Review Journal ofApplied Mechanics 46, 241–258.

Sarpkaya, T., 1995. Hydrodynamic damping, flow-induced oscillations, and bihar-monic response. Journal of Offshore Mechanics and Arctic Engineering 117,232–238.

Singh, S.P, Mittal, S., 2005. Vortex-induced oscillations at low Reynolds numbers:hysteresis and vortex-shedding modes. Journal of Fluids and Structures 20,1085–1104.

M. Zhao et al. / Ocean Engineering 73 (2013) 83–9594

Sumer, B.M., Fredsøe, J., 1988. Transverse vibrations of an elastically mountedcylinder exposed to an oscillating flow. Journal of Offshore Mechanics andArctic Engineering 110, 387–394.

Sumer, B.M., Fredsøe, J., Jensen, K., 1994. A note on spanwise correlation on a freelyvibrating cylinder in oscillatory flow. Journal of Fluids and Structures 8,231–238.

Sumer, B.M., Fredsøe, J., 1997. Hydrodynamics Around Cylindrical Structures. WorldScientific, Singapore.

Yang, B., Gao, F.P., Wu, Y., 2006. Experimental study on vortex-induced vibrations ofsubmarine pipeline near seabed boundary in ocean currents. China OceanEngineering 20 (1), 113–121.

Yang, B., Gao, F.P., Jeng, D.S., Wu, Y.X., 2008. Experimental study of vortex-inducedvibrations of a pipeline near an erodible sandy seabed. Ocean Engineering 35,301–309.

Williamson, C.H.K., 1985. Evolution of a single wake behind a pair of bluff bodies.Journal of Fluid Mechanics 159, 1–18.

Zhao, M., Cheng, L., 2011. Numerical simulation of two-degree-of-freedom vortexinduced vibration of a circular cylinder close to a plane boundary. Journal ofFluids and Structures 27, 1097–1110.

Zhao, M., Cheng, L., An., H., 2012. Numerical investigation of vortex-inducedvibration of a circular cylinder in transverse direction in oscillatory flow.Journal of Ocean Engineering 41, 39–52.

Zhao, M., Cheng, L., Zhou, T., 2011. Numerical investigation of vortex-induced (VIV)vibration of a circular cylinder in oscillatory flow. In: Proceedings of the 30thInternational Conference on Ocean, Offshore and Arctic Engineering,OMAE2011 June 19–24, 2011, Rotterdam, The Netherlands.

Zhao, M., Cheng, L., Teng, B., Dong, G., 2007. Hydrodynamic forces on dual cylindersof different diameters in steady currents. Journal of Fluids and Structures 23,59–83.

Zhao, M., Cheng, L., Zhou, T., 2013. Numerical simulation of vortex-inducedvibration of a square cylinder at a low Reynolds number. Physics of Fluids 25(2) (023603).

M. Zhao et al. / Ocean Engineering 73 (2013) 83–95 95