Numerical investigation of wake induced vibrations of cylinders in tandem arrangement at subcritical Reynolds numbers

Ocean Engineering 154 (2018) 341–356

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Numerical investigation of wake induced vibrations of cylinders in tandem arrangement at subcritical Reynolds numbers Vinh-Tan Nguyen *, Wai Hong Ronald Chan, Hoang Huy Nguyen Institute of High Performance Computing, Agency for Science, Technology and Research, Singapore (A*STAR), 1 Fusionopolis Way, #16-16 Connexis, 138632, Singapore

A R T I C L E I N F O

A B S T R A C T

Keywords: Flow induced vibrations Vortex induced vibrations Computational fluid dynamics Detached eddy simulations High Reynolds number flows Fluid-structure interactions

Wake-induced vibrations are considered distinctively different from well-known vortex induced vibrations where bluff body structures vibrate under vortex shedding in the wake of flows passing through them. Understanding characteristics of wake-induced vibrations plays an important role in design of marine-offshore structures such as platform columns, risers operating at high Reynolds number currents. This work presents an extensive study of wake-induced vibrations using a computational fluid dynamics approach. The numerical model is based on an incompressible Navier-Stokes flow solver with a hybrid detached eddy simulation (DES) approach for turbulence modelling of high Reynold number flows. The hybrid DES approach is first validated for wake-induced vibrations at subcritical Reynolds number ranging from 103
105 . Comparison between numerical results and experiment shows very good agreement in prediction of downstream cylinder responses including amplitudes, frequency and phase angle. Data from numerical simulations is then used to characterize and study mechanisms of wake-induced vibrations by looking at detailed force components and their frequencies in relation to cylinder responses. The effect of different Reynolds number flow conditions in tandem cylinder responses are also analysed in the present work. It is found that Reynolds number has strong influence in changing response amplitude and frequency of cylinders in wake-induced vibrations.

1. Introduction Understanding of environment loads due to waves and currents is important in designing offshore structures such as cables, risers, mooring line systems. Apart from hydrostatic pressure forces, offshore structures under cross currents are normally subjected to dynamic load due to the well-known vortex shedding phenomena. Coupled interaction of vortex shedding from bluff bodies with its structural flexibility causes vibration responses of the structure. This vortex-induced vibration (VIV) is widely observed in various types of cylindrical structures in oceans, heat exchanger tubes, as well as chimney stacks, among others. These vibrations have been extensively studied in the literature (Williamson and Govardhan, 2004), and are modelled as resonant vibrations of elastically mounted and isolated rigid cylinders. Vibrations associated with the VIV phenomenon are known to have large amplitudes in an extended “lock-in” frequency region, where the response frequency f is maintained at near the natural frequency fN of the structure over a range of driving frequencies fF . In most of offshore installations including deep sea oil-rig platforms, risers are arranged in bundles and arrays. Under array arrangement risers

and similar cylindrical structures under cross flows may not only be affected by VIV but they also interfere with each other. It has been observed and noted (Paidoussis and Price, 1988) that arrays of cylinders are further subject to fluid-elastic instabilities that arise from interactions between adjacent cylinders in the array. Also termed wake-induced vibrations (WIV), these instabilities have been recognized as a distinct phenomenon from VIV and have been investigated extensively as well (Zdravkovich, 1977; Bokaian and Geoola, 1984; Brika and Laneville, 1999; Mittal and Kumar, 2001; Borazjani and Sotiropoulos, 2009; Assi et al., 2010a, 2010b; Huera-Huarte and Bearman, 2011; Assi et al., 2013; Chaplin and Batten, 2014; Assi, 2014). However, to date, no coherent theory has emerged on the WIV phenomenon, even though analyses involving nonlinear oscillators (Facchinetti et al., 2002; Armin and Srinil, 2013; Gallardo et al., 2014) and wake stiffness (Assi et al., 2010b, 2013; Assi, 2014) have been proposed. In particular, it has been suggested that WIV is not a resonant phenomenon, and that the response and driving frequencies are equivalent (Assi et al., 2010b, 2013; Assi, 2014). In addition, due to the coupling between the motion of the structure and the flow of the surrounding fluid, particularly the generation of vortices from the interaction between the fluid and the structure, it is conceivable that

* Corresponding author. E-mail address: [email protected] (V.-T. Nguyen). https://doi.org/10.1016/j.oceaneng.2018.01.073 Received 8 November 2016; Received in revised form 5 May 2017; Accepted 16 January 2018 0029-8018/© 2018 Elsevier Ltd. All rights reserved.

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Fig. 1. Illustration of the physical setup in our model of tandem cylinders.

Table 1 List of numerical experiment conducted in the present work and their corresponding references if available. F1 is the case of fixed cylinders and all other cases are for studying of wake-induced vibrations. X/D

m

ζN

Reference

4

–

–

Ljungkrona et al. (Ljungkrona et al., 1991)

3:04  104

4.75

3.0

0.04

0:2
3:0  104 1:0  104 1:0  104

5.0

2.6

0.007

Hover and Triantafyllou (Hover and Triantafyllou, 2001) Assi et al. (Assi et al., 2010b)

4.75 5.0

3.0 2.6

0.04 0.007

– –

Cases

Re

F1

2:0  10

W1 W2 W3 W4

4

the frequency decomposition of the driving force is time-varying and possibly a harmonic. Majority of earlier work on wake-induced vibrations are physical experiment listed above including many scaled model as well as full scale tests. While computational fluid dynamics (CFD) has become popular in engineering applications over the past few decades, its application in modelling and simulations of wake-induced vibrations is still limited and lacking behind experimental work. Among those limitations of CFD models for wake-induced vibration simulations is its predictive capabilities at high Reynolds number flows in the range of Re ¼ 104
106 corresponding to operational conditions of most offshore structures. Direct numerical simulations (DNS) have been attempted for vortex induced vibrations at Re ¼ 104 (Dong et al., 2006); however at the range of higher Reynolds number conditions, DNS becomes inhibited due to unrealistic computational demand even with present powerful computing infrastructures. Significance of resolving flows past cylinders at high Reynolds number has increasingly motivated the CFD community to explore various advanced turbulent modelling techniques from traditional industrial unsteady Reynolds average Navier-Stokes (URANS) (Rosetti et al., 2012) models to large eddy simulations (LES) (Feymark et al., 2012). In RANS all turbulence is modelled and it solves for ensemble average quantities. By separating flows to mean and fluctuating parts, RANS models normally require some empirical approximation to model turbulent fluctuations. In LES, larger turbulent scales are resolved while small scales called subgrid scales (SGS) are modelled. A spatial filtering is then employed to separate the resolved turbulence from its modelled counterpart. For accurate LES predictions, it requires a suitable filtering mechanism to separate large wave components from smaller ones at a cut-off frequency. Moreover it also restricts the time step and grid sizes, especially for wall bounded flows. As an attempt to strike a balance between resolving and modelling, detached eddy simulations DES (Travin et al., 2000; Spalart, 2009) was introduced as a hybrid RANS/LES approach. One can refer to those work as well as references cited therein for more details on the applications of various turbulence models for simulations of flow induced vibrations. In this work, a fluid-structure interactions numerical model is

Fig. 2. Domain set up and typical grid for the current simulations of wakeinduced vibration. (a) A computational domain with boundary conditions and the focus region bounded by the dash lines. A typical mesh topology used for the simulations with isotropic mesh refinemenet in the focus region. Boundary layer regions are generated around the cylinders with a-priori estimated of y þ  1.

Fig. 3. Instantaneous isosurface of Q ¼ 1 coloured by vorticity component in spanwise direction for flows over tandem cylinder at X=D ¼ 4 and Re ¼ 2  104 . It shows complex three-dimensional wake structures resulted from upstream wake interacting with downstream cylinder.

presented as an alternative tool for investigation of wake-induced vibrations. Here, a fluid flow solver based on a DES approach (Nguyen et al., 2015) for turbulence modelling at high Reynolds number flows was used for simulations of wake-induced vibrations. The main objectives of the current work is two-fold. It is first to validate the FSI model through extensive benchmarking with experimental studies in prediction of WIV responses of tandem cylinders at the same flow conditions with physical experiment set-ups. The numerical model is capable of predicting WIV responses with pronounced agreement to earlier experimental data; thus making it a suitable alternative tool for further

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Fig. 4. Contour plots of velocity magnitude as well as ratio of LES and RANS length scale at the center cut-plane z ¼ 0. In (b), the dark region in the wake shows where LES is active as expected from the hybrid scheme. It is also noted that RANS is active in boundary layer regions as well as far-field.

Table 2 Mesh convergence study for flows over tandem cylinder arrangement with X=D ¼ 4. The mesh were refined by reducing stretching ratio (ΔS ) between focus region mesh size and the first bounday layer thickness while keeping the y þ  1. Grids

Upstream Cyl

ΔS

No. of Cells

1 2 3 4

60 50 40 30

653 324 1 004 928 2 055 552 4 512 620

Exp (Ljungkrona et al., 1991) Error (%)

– –

– –

Downstream Cyl

Strouhal

GCI(%)

1.418 0.801 0.836 0.849

0.1768 0.1863 0.1909 0.1905

– 23.16 2.67 0.40

– –

0.1840 3.75%

Cd

C'l

Cd

C'l

1.406 1.104 1.058 1.069

0.997 0.426 0.339 0.365

0.485 0.520 0.540 0.543

1.195 10.54%

– –

0.4281 26.8%

Fig. 5. Comparison of mean and fluctuation of pressure coefficients on the surface of downstream cylinder at Re ¼ 2  104 , X/D ¼ 4. The results from LES (Kitagawa and Ohta, 2008) are at higher Reynolds number Re ¼ 2:2  104 .

set-up are presented. This includes details of numerical methodologies as well as grid generation, boundary conditions. In section 3, we present the validation of the model with earlier experimental studies to establish model's fidelity for subsequent analysis of forces and responses of wake-induced vibrations. The following section 4 discuses a study of Reynolds number effects on WIV. Different from most of physical experimental set-up, Reynolds number in the numerical study is maintained for variation of reduced velocities to decouple their effects on WIV responses. The effect of Reynolds number conditions on

study of WIV phenomena. Secondly it is to demonstrate the application of the developed model for investigation of wake-induced vibrations in complementing physical experiment. In particular, the numerical model is capable of providing more detailed analysis of flow features while allowing more control in setting up the experiment. It is with this expectation that the numerical model can further advance physical understanding of wake interactions on tandem cylinder's responses. The remaining of the paper is organized as follows. In the next section, a brief introduction about the FSI model and numerical experiment's 343

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Fig. 6. Force coefficients on upstream and downstream cylinders in tandem arrangement with X/D ¼ 4 and Re ¼ 2  104 .

Table 3 Comparison of force coefficients and frequency between present model with experimental data and LES at different Reynolds numbers. Reynolds

Upstream Cyl

Downstream Cyl

Strouhal

Cd

C'd

C'l

Cd

C'd

C'l

Exp. (Ljungkrona et al., 1991) Exp. (Alam et al., 2003) LES (Kitagawa and Ohta, 2008)

2:0  104 6:5  104 2:2  104

1.195 1.217. 1.185

– 0.1025 0.0804

– 0.4576 0.4693

0.4281 0.3691 0.3430

– 0.1878 0.4528

– 0.7742 1.2477

0.1840 0.1917 0.1890

DES -

2:0  104 6:5  104

1.0584 1.3461

0.0901 0.0951

0.3387 0.5448

0.5401 0.5001

0.1336 0.17438

0.8357 0.9420

0.1909 0.1859

Fig. 7. Spectrum of velocity fluctuation and forces on downstream cylinder at X/D ¼ 4: (a) power spectrum of velocity fluctuation measured at two location in the wake of upstream and downstream cylinder; (b) lift force coefficient of downstream and upstream cylinder showing single Strouhal frequency response.

different WIV tests are investigated in details to provide insights on characteristics and mechanism of WIV responses. Some conclusion remarks from the current study are presented in the last section.

was constructed in the numerical model similar to many earlier experimental studies. Here we consider effects of a long slender cylinder (cylinder 1) on vibrations of an elastically mounted and parallel downstream cylinder of the same length L (cylinder 2), as briefly illustrated in Fig. 1. Generally cylinder 1 (with diameter D1 ) is fixed to a rigid frame while cylinder 2 (with diameter D2 ) is mounted to the rigid frame with effective spring constant k and effective damping constant c. Structural responses of the downstream cylinder are characterized by its natural frequency and damping ratio defined as follows

2. Model and methodology 2.1. Physical setup In this work a canonical setup of cylinders in tandem arrangement 344

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Fig. 8. Lift coefficients history and their spectrum on upstream cylinder 1 and downstream cylinder 2 obtained from numerical simulations at Re ¼ 19200, Y0 =D ¼ 1, X0 =D ¼ 4.

Fig. 9. Wake structures for cylinder in tandem at Re ¼ 19200, Y0 =D ¼ 1, X0 =D ¼ 4 corresponding to (a) maximum lift and (b) minimum lift on the downstream cylinder.

ωN ¼

rffiffiffiffi k c ; ζ ¼ pffiffiffiffiffiffi : m N 2 km

vibrations which in turns change vortex shedding modes and flow dynamics. Generally frequency of vortex shedding from a cylinder can be defined as

(1)

Here m is the effective mass of the downstream cylinder. Taking the direction of the cylinder axes to be the z-axis and the downstream direction to be the x-axis, we further define cylinder 2 to be of distance X0 downstream of cylinder 1 in the x-direction and distance Y0 above cylinder 1 in the y-direction. In the present work we consider the two cylinders are of the same diameters D1 ¼ D2  D . The problem is further simplified by restricting the motion of the downstream cylinder 2 to the y-direction and call its instantaneous displacement from the equilibrium position b y as previously studied by various authors (Zdravkovich, 1977; Bokaian and Geoola, 1984; Hover and Triantafyllou, 2001; Assi et al., 2010b). We assume that both cylinders are completely immersed in a Newtonian fluid with density ρ and dynamic viscosity μ. In addition, the fluid is moving uniformly in the x-direction and has a speed of U∞ far away from the cylinders. Under cross flows conditions characterized by Reynolds number ReD ¼ ρU∞ D=μ, vortex shedding from cylinders result in force fluctuations acting on the cylinder causing

fV ¼

StU∞ D

(2)

where St is the dimensionless Strouhal number associated with the flow and bluff body geometry, U∞ is the characteristic in-flow velocity. In addition, we define the reduced velocity Ur ¼

U∞ ; Df0

(3)

which is another dimensionless number describing the ratio of the characteristic in-flow velocity to the characteristic cross-flow velocity scale. Here f0 is the natural frequency of the downstream cylinder. In this work, we consider an in-line arrangement where both cylinders lie on the x-axis in a tandem arrangement; giving Y0 ¼ 0. For tandem arrangement between the cylinders, the spacing distance X0 is 345

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Fig. 10. Wake induced vibrations of cylinder in tandem with X0 =D ¼ 5D1 =D2 ¼ 1, Ur ¼ 15. (a) Displacement and lift force coefficient on downstream cylinder plot over repeated periods of oscillation and (b) their FFT showing single displacement frequency under multiple frequency of lift force.

Fig. 11. Plot of mean dimensionless displacement amplitude against reduced velocity for X0 =D ¼ 5 following the same experimental conditions by (Assi et al., 2010b) (m ¼ 2:6, m ξ ¼ 0:018, Re  25000). The results are also plotted with recent experiment by (Chaplin and Batten, 2014) (m ¼ 0:87, m ξ ¼ 0:025, Re  58000). Present CFD results on displacement amplitude agree well with experimental data over a range of reduced velocity.

Fig. 12. Comparison between present CFD results on displacement amplitude with experimental data by (Assi et al., 2010b) for different separation distances between cylinders. The solid symbols are the present CFD results.

2.2. Fluid structure interaction computational model We propose to use a strong coupled fluid-structured interaction (FSI) computational model to investigate response of cylinders in tandem arrangement under different conditions. The numerical model based on our earlier work (Nguyen and Nguyen, 2016) strongly couples a hybrid detached eddy simulation (DES) flow solver with six degree of freedoms (6DOF) solver for solving FSI response of downstream cylinders under cross flows and wakes from upstream cylinder. Here flows over cylinders are modelled as viscous incompressible fluids governed by the incompressible Navier-Stokes equations that express the conservation of mass and momentum. Using Reynolds decomposition, the governing Navier-Stokes equations can be written as follows

taken to be sufficiently large that vortices develop between the two cylinders. This excludes the scenario of near-wake interference considered by some authors (Borazjani and Sotiropoulos, 2009; Huera-Huarte and Bearman, 2011). Responses of downstream cylinders in tandem arrangement can be written as a function of non-dimensional numbers (Assi et al., 2010b) by ¼G D

  X0 f ReD ; St; ; ; ζN ; m : D f0

(4)

Here f denotes response frequency of the downstream cylinder. And m is the mass ratio measuring the ratio of cylinder mass to the displaced fluid mass. It is clear that multiple factors influence characteristics and responses of a cylinder in a wake of another upstream cylinder. In this work we first focused on validating the numerical models with experimental data at selected flow regimes and physical parameters. Subsequently a detailed analysis of flows at different Reynolds numbers were conducted to understand its effects on WIV responses. Table 1 summarizes different conditions considered in this work.

r⋅u ¼ 0

(5)

∂u 1 1 þ r⋅ðuuÞ ¼
rp þ νr2 u
r⋅τ R ∂t ρ ρ

(6)

In these equations, velocity vector is denoted by u, ρ is the density, p is the pressure and ν is the kinemtaic viscosity of the fluid. The overbar denotes time-averages of the variables. In the present work, water is 346

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Fig. 13. Mean displacement and lift force over one cycle of downstream cylinder oscillation at various reduced velocity conditions at X0 =D ¼ 5, following experiment setup in (Assi et al., 2010b). In those figures, light colour lines are instantaneous history of displacement and lift coefficients over each cycle; bold colour lines are their corresponding mean values over a cycle. Solid lines are displacement amplitude and dotted lines are lift coefficient. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

considered as Newtonian fluid of density ρ ¼ 1000 kg m
3 and kinematic viscosity of ν ¼ 1:0  10
6 m2 s
1 . The last term on the above momentum equation is normally referred to as the Reynolds stress tensor τ R ¼ ρðu'u'Þ. The system is closed with appropriate boundary conditions imposed on the boundary of the domain Γ ¼ δΩ u¼u

in ΓD ;

σ ⋅n ¼ t in ΓN ;

scale is no longer a pure LES length scale but based on the subgrid-scale turbulent kinetic energy and dissipation rate. In order to prevent the grid induced separation in pure DES form due to fine grid resolution near wall (Spalart, 2009) causing unsteadiness in the boundary layer, the DES length scale is modified to disable LES in boundary layer region using the model blending function. The system of governing equations is then solved using the unstructured collocated finite volume method in arbitrary Lagragian-Eulerian framework with pressure implicit with splitting operators (PISO) for pressure-velocity coupling. Responses of the downstream cylinder in transverse direction represented as spring-mass-damper system are governed by the equation of motion written as

(7) (8)

where domain boundary is separated as Γ ¼ ΓN [ ΓD denoting a Neumann and Dirichlet boundary respectively. More detailed boundary conditions for the simulations are described in the next section. In this work a hybrid RANS-LES model is employed for turbulence modelling. For hybrid RANS/LES approach, a DES formulation is implemented and based on Menter's k
ω SST model (Strelets, 2001) and later modified in (Menter and Kuntz, 2004) by making the RANS model become length scale aware. In this implementation, in the LES mode the turbulent length

€y þ c2 by_ þ k2 by ¼ Fy : m2 b

(9)

Here b y is the transverse cylinder displacement and Fy is the force component in y direction. The force acting on the cylinder is evaluated by integrating pressure and viscous force obtained from the Navier-Stokes flow solver. 347

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Fig. 14. Spectrum of force and lift force coefficient on downstream cylinder at various reduced velocity conditions at X0 =D ¼ 5, following experiment setup in (Assi et al., 2010b). Here frequencies are normalized with the natural frequency of the downstream cylinder, f0 . Solid lines are response amplitude spectrum and dotted lines are power spectrum of lift coefficient.

Fy ¼ ∫ ey ⋅ðpI þ τ Þnds

boundary and pressure outlet condition was applied in the downstream boundary. On the cylinder walls no-slip conditions were specified while slip conditions are for the top and bottom boundaries. Hybrid RANS-LES simulations require strict mesh grading of boundary layer mesh with y þ  1 while sufficient mesh resolution in the wake to enable LES model. Here hexahedral dominated grids with topology shown in Fig. 2(b) are generated with higher resolution closer to the cylinders to save computational cost. The focus region is typically about 20  10D in most of the simulations. Flows in the current range of Reynolds number are inherently three-dimensional and the span size of cylinder strongly affects the flow dynamics, especially lift fluctuations. It was recommended that an aspect ratio of L=D ¼ 9
12 is required to fully capture the threedimensional flow features (Norberg, 2003). From our earlier study (Nguyen and Nguyen, 2016) the aspect ratio of L=D ¼ 9 was used in all of the current simulations except otherwise specified. A hexahedral dominated mesh is generated for the above computational domain using snappyHexMesh tool in OpenFOAM. For hybrid RANS-LES simulations, extra care is required for mesh generation to avoid ambiguous grid densities (Spalart et al., 2006). In this work the near wall boundary layer

(10)

Γcyl

Here I is the identity tensor and ey is the unit vector in y-direction. The equation of motion is discretized in time using Leapfrog scheme and coupled with the flow solver in each of the pressure-velocity coupling outer corrector iteration; thus rendering a strong coupling approach for the fluid structure interaction system. For detailed description of the hybrid DES model used in this work one can refer to the earlier work (Nguyen and Nguyen, 2016) and references cited therein. 2.3. Computational grid and numerical schemes The computational domain and typical mesh topology for all simulations of flows over tandem cylinder arrangement in the present work are shown in Fig. 2. The domain is extended to 15D in upstream and 30D in the wake direction from the downstream cylinder where D is the diameter of the downstream cylinder. In the cross flow direction typically it is extended to about 20D depending on the blockage ratio used in experimental set-ups. Velocity inlet condition was specified at the inlet 348

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equations were solved by the smooth solver in each step until a convergence tolerance was met. The convergence criterion was set to 10
8 for turbulence variables and 10
6 for other quantities to give reasonably good convergent results from prior experience. Second order implicit backward scheme is employed to advance the system in time with maximum CFL number of 0.5–0.75 for all of the simulations. For handling 6-DOF motions of cylinder under wake-induced vibration, a dynamic morphing mesh approach was used to deform computational meshes where mesh motion is interpolated from cylinder's motions using spherical linear interpolation. Using this approach the mesh is deformed according to cylinder motions while maintaining its smoothness and preserving mesh quality in boundary layer regions of solid body motions. 3. Numerical results and discussion 3.1. Verification and validation for fixed cylinders in tandem arrangement While the model had been well-validated for vortex induced vibrations of single cylinders at sub-critical Reynolds numbers in our previous work (Nguyen and Nguyen, 2016), it was again further validated for the present setting of tandem cylinder arrangement. There are numerous experimental work in literature on investigation of flows over tandem cylinders (Igarashi, 1981; Ljungkrona et al., 1991; Alam et al., 2003) including a comprehensive review (Sumner, 2010) and many other references cited therein. A similar set-up of cylinders in tandem as in (Ljungkrona et al., 1991) was chosen for benchmarking of the current model. The set-up comprises two cylinders with the same diameter of D ¼ 20mm arranged in-line with separation distance of X=D ¼ 4 and blockage ratio of 6:3%. The cylinders were under flows of Reynolds number Re ¼ 2  104 with inlet turbulent intensity of Tu ¼ 0:1%. A similar set-up was also used in recent work (Kitagawa and Ohta, 2008) for validation of large-eddy simulations (LES) for flows over tandem cylinders with slightly higher Reynolds number of Re ¼ 2:2  104 . Fig. 3 shows an instantaneous profile of three dimensional wake structures for the tandem cylinder arrangement. In more details, Fig. 4 depicts flow velocity contour at the center cut plane of the domain and the DES indicator contour. Here the DES indicator is measured as the ratio of the length scale between LES and RANS. From Fig. 4b, the dark region in the wake is the active LES region while the RANS model is only activated in the boundary layer regions around the cylinders as well as the far-field. This has demonstrated that the hybrid RANS/LES model works as expected to resolve the wake flows with LES while RANS is employed to model wall bounded flows. It is believed that resolving the wakes appropriately well and their interaction with the downstream cylinder provides more accurate computations of forces acting on the cylinder; thus giving good prediction of wake-induced vibrations. Table 2 summarizes results of the force coefficients on the upstream and downstream cylinders for different mesh sizes. The meshes were refined by reducing the stretching ratio between the cell size in the focus region and the first boundary layer thickness which is kept at a constant value of δy1 ¼ 1:58  10
4 m to maintain y þ  1 at this Reynolds number. In the current work, Cd denotes the mean of drag force coefficient and C'l is the root mean squared (r.m.s) lift coefficient measuring fluctuation of lift forces on the cylinders. As the mesh was refined from Δs ¼ 60 to 30, the mean drag coefficient and lift fluctuation on the upstream and downstream cylinder approached a converged value. The grid convergence index (GCI) was computed from the mean drag coefficient of the downstream cylinder on the two consecutive grids and shown in the table. As the grid was refined from level 3 to 4 the GCI value is less than 1%. It shows that the numerical prediction is approaching the asymptotic numerical value. In our subsequent simulations grid level 3 or equivalent at different Reynolds numbers was employed to sufficiently guarantee numerical grid convergence. It is noted that there is a discrepancy between asymptotic numerical prediction and experimental data, especially for the mean drag coefficient on the downstream

Fig. 15. (a) Plot of response frequency normalized with natural frequency against reduced velocity and (b) phase difference between response and lift against reduced velocity for DX02 ¼ 5. It is noted that the response frequency is increased with increase in reduced velocity. Comparison between CFD and experimental results (Chaplin and Batten, 2014; Assi et al., 2010b) show good agreement.

mesh resolution is always maintained a y þ  1 for a given Reynolds number flow regime. The first layer thickness (δy1 ) can be estimated þ

a-priori based on y þ and a friction velocity u as δy1 ¼ ρy uμ in which

friction velocity is computed from a skin-friction formula for flatplate at the specified Reynolds number. From the first boundary layer thickness the mesh resolution in the focus region can be determined based on a user Δx ); thus providing number of necessary defined stretching ratio(ΔS ¼ δy 1 refinement levels in this region. It is worth noticing that the mesh is not further stretched in spanwise direction as isotropic refinement is applied in the computational domain. Flows over cylinders at this range of Reynolds number exhibits highly unsteady characteristics and were simulated using the transient dynamic mesh motion solver implemented in OpenFOAM framework (Nguyen and Nguyen, 2016). Similarly to our earlier work, a second order linear interpolation scheme was used for convective terms while a cell limiting scheme determining the limited gradient along a line connecting adjacent cell centres was used for the discretization of diffusive terms. Non-orthogonal and skewness correctors were also employed to maintain second order accuracy on unstructured meshes. For all the simulations in this work, the linear solvers of geometric-algebraic multi-grid solvers (GAMG) and preconditioned bi-conjugate gradient (PBiCG) were used to solve the pressure and velocity respectively. In addition, the turbulence 349

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Fig. 16. Comparison of response amplitude and frequency at X=D ¼ 5 with different Reynolds number conditions. Results from variable Reynolds number following experimental setup (Assi et al., 2010b) and present CFD simulations with constant Re ¼ 104 shows the effects of Reynolds number on wake-induced vibrations.

Fig. 17. Force and amplitude response spectrum of cylinder in tandem (X=D ¼ 5) at the same reduced velocity but different Reynolds number.

number Re ¼ 2  104 . For downstream cylinder the present results show higher mean drag and lower fluctuations as compared to experimental data. Given uncertainties in force measurement on the cylinder due to various factors such as aspect ratio and blocking ratio, the numerical prediction for force coefficients are consistently well within the range of data reported in literatures (Sumner, 2010). In Fig. 7(a), power spectra of cross flow velocity are plotted at a point in between the separation between two cylinders and a point in the wake of the downstream cylinder. The Strouhal frequency was clearly shown in the flow spectra for both points. Higher frequency peaks are also clearly present more in velocity spectrum in the wake of downstream cylinder probably due to interaction of upstream wakes with downstream cylinder shear layers. However a single frequency response is observed from force spectrum of lift coefficients on both upstream and downstream cylinders as shown in Fig. 7(b). Fig. 8 depicted the variation of force coefficients on the cylinders for a fixed arrangement of X0 =D ¼ 4 awith the offset distance of Y0 =D ¼ 1, similar to one of the conditions tested in the earlier experiment (Assi

cylinder. Pressure distribution on downstream cylinder is plotted in Fig. 5 along the circumference of cylinder. The mean and root mean square p
p0 fluctuation of pressure coefficients (Cp ¼ 1=2 ρU 2 ) are compared between ∞

the present computation with experimental data as well as LES results obtained from the similar Reynolds number in good agreement. It can be seen that the mean pressure coefficient tends to flatten out in the back of the downstream cylinder with lower fluctuation as compared to the front part which is in direct impingement of upstream wakes. This results in much higher lift fluctuation and lower drag on the downstream cylinder as shown in Fig. 6. Force coefficients Cl;d ¼ Fl;d =   1 2 on the upstream and downstream cylinder are compared ρ LDU ∞ 2 with experimental data as depicted in Table 3. The current numerical model provides lower predictions of force coefficient for upstream cylinder as compared to experiment as well as LES at the Reynolds

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Fig. 18. Force frequency on cylinders for constant and varying Reynolds number for tandem arrangement (X=D ¼ 5). Each line in the figure shows the lift force spectrum at a reduced velocity. It is noted that lift force on upstream cylinder shows single frequency while the downstream cylinder has two dominant frequencies in lift force spectrum.

lift force acting on the downstream cylinder.

et al., 2010b). Different to the above in-line tandem arrangement, the mean lift coefficient of the downstream cylinder is negative in this case and it acts as a restoring force to pull the cylinder towards the centerline. This is believe is one of the key mechanism explaining wake-induced vibrations. This observation is consistent with the earlier experimental data. It can also be seen that the fluctuation of lift force coefficient on the downstream cylinder is much higher than the upstream's one while the force frequency is the same on both cylinders corresponding to Strouhal number of around St ¼ 0:2. The increment in lift force is due to interactions of upstream cylinder wakes with the downstream cylinder showing effects of vortices from the upstream cylinder on the downstream one. In Fig. 9, wake structures of the cylinders in tandem are plotted at the maximum and minimum lift force conditions. At the maximum lift condition in Fig. 9(a), upstream vortices induce a high speed flow at the bottom side of the cylinder and accelerate a shear layer from the same side; thus increasing lift force on the downstream cylinder. When the upstream vortices impinge on the downstream cylinder as shown in Fig. 9(b), it interact with the shedded vortex from downstream cylinder and resulted in a smaller lift force. A similar wake structures was also observed in the experiment (Assi et al., 2010b) where a complex vortex-structure interaction resulted in enhancing and diminishing the

3.2. Validation for wake induced vibration cylinders in tandem arrangement We next consider a case of a vibrating cylinder in a tandem arrangement with an upstream cylinder and zero transvered offset distance. The test case follows the same experimental setup in (Assi et al., 2010b) in which a cylinder of distance X0 =D ¼ 5 from upstream and fixed cylinder is free to move in cross flow direction under different reduced velocities. In the numerical study, the downstream cylinder is modelled as six degree-of-freedom rigid body with the same mass ratio m ¼ 2:6 and structural damping of ζ ¼ 0:7% as in the experiment. Here the cylinder was constrained to only oscillate in the cross flow direction. The history of displacement and lift force coefficients of the downstream cylinder was plotted in Fig. 10 for reduced velocity of Ur ¼ 15, corresponding to Reynolds number of Re ¼ 1:5  104 . It can be seen that the downstream cylinder oscillated with a single frequency close to its natural frequency while power spectrum of lift coefficient shows multiple frequency components in the driving force acting on the cylinder. Comparison of response amplitude between present numerical sim351

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Fig. 19. Comparison of response amplitude at Triantafyllou, 2001).

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X D

¼ 4:75 with experimental data at the same condition of constant Reynolds number Re ¼ 3:05  104 (Hover and

Fig. 20. Response amplitude and phase difference at different Reynolds numbers of Re ¼ 104 (X/D ¼ 5) and Re ¼ 3:05  104 (X/D ¼ 4.75). The CFD results are plotted with experimental data at the same condition of constant Reynolds number Re ¼ 3:05  104 (Hover and Triantafyllou, 2001).

experimental data extracted from the earlier work (Assi et al., 2010b; Chaplin and Batten, 2014) in prediction of wake-induced vibration response for cylinder in tandem arrangement.

ulations and experimental study is shown in Fig. 11 for downstream cylinder at different reduced velocities. It is noted from Fig. 11 that displacement amplitude increases with increase of reduced velocity for the cylinder in tandem. This is a distinct behaviour of wake-induced vibrations as compared to single cylinder vortex induced vibrations. In VIV, response amplitude is large only in a range of reduced velocity where natural frequency matches with vortex shedding frequency; away from this locked-in region cylinder response amplitude is negligibly small. For cylinders in wake-induced vibrations, the response is similar to VIV in the region of low reduced velocity (Ur < 8) while wakes from the upstream cylinder played a critical role in downstream cylinder's response at higher reduced velocity; thus causing high amplitude responses. Fig. 12 showed amplitude response of downstream cylinder with reduced velocities at different spacing between two cylinders. Numerical results showed the same trend as experiment data (Assi et al., 2010b) where response amplitude expectedly reduces with spacing distance. It is clear that the present numerical results generally agrees well with the

3.3. Force and displacement of cylinders in tandem Distinctive responses of cylinder in tandem at high reduced velocity as shown in Fig. 11 have been attributed to complex interactions of cylinder wakes and its motions (Assi et al., 2010b; Chaplin and Batten, 2014). Outside of VIV lock-in regions, wakes from upstream cylinder interacts with downstream cylinder's vortices causing downstream cylinder vibrations. Amplitude of downstream cylinder motions at high reduced velocity is dependent on distance between the two cylinders as shown in earlier work (Assi et al., 2010b) and the current numerical results are in good agreement with the experimental data in Fig. 12. The response displacement of downstream cylinder was plotted against its lift force coefficient at different reduced velocities in Fig. 13. Overlaying the 352

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Fig. 21. Response frequency of cylinder in tandem arrangement at different Reynolds numbers.

Fig. 22. Mean drag coefficient and root-mean-square of lift force coefficient on downstream cylinder in tandem arrangement at different Reynolds numbers.

displacement was observed at Ur ¼ 6 similar to VIV phenomena and it is noted that phase difference is measured only for the force component at the same frequency with response frequency. The plots clearly showed that our results corroborate well with those obtained from experimental data (Assi et al., 2010b; Chaplin and Batten, 2014). It is well understood that WIV responses of cylinders in tandem depends on many parameters including flow conditions as well as geometric configurations as illustrated in Eq. (4). Coupled effects of those parameters results in complex and distinctive behaviours of wakeinduced vibrations responses. With pronounced agreement in comparison to experimental data, the present numerical model is capable of resolving key flow physics involved in wake-induced vibrations. Additionally it provides an alternative analysis tool to further investigate mechanisms of wake-induced vibrations. In the next sections the numerical model is employed to further characterize as well as to study the effects of Reynolds number on responses of cylinders in wake-induced motions.

data over one cycle of displacement shows multiple frequency modes in lift force coefficient. The difference in phase and frequency between mean lift coefficient and mean displacement is clearly observed for all reduced velocities. To further illustrate this characteristic of downstream cylinder in tandem, Fig. 14 shows displacement spectral density as well as lift force spectrum of the downstream cylinder at different reduced velocities. It is noted that the magnitude of displacement measured in root mean square gets higher with increase of reduced velocity. As seen from the power spectral density plots, the displacement response is at a single frequency while there are consistently two dominant frequencies in lift force component of the downstream cylinder at various reduced velocities. One of the two dominant frequencies is the same as motion frequency of the downstream cylinder while the other frequency matches with vortex shedding frequency from upstream cylinder. The presence of second frequency in lift force power spectrum demonstrates the effect of upstream wakes on the downstream cylinder. Response frequency of the downstream cylinder and the phase angle between the cylinder's lift coefficient and displacement were plotted in Fig. 15 for the same case of X=D ¼ 5. It is found that the response frequency of the cylinder increases with reduced velocity and consistently matching with earlier experimental data. Moreover the response frequency of cylinder in WIV shown in Fig. 15 increasingly deviates from its natural frequency with increase of reduced velocity. A shift in phase angle between force and

3.4. Effect of Reynolds number on wake induced vibrations In most of the studies on wake-induced vibrations (Assi et al., 2010b; Assi, 2014; Chaplin and Batten, 2014) responses of the downstream cylinder are characterized with variation of reduced velocity while flow 353

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velocity by deviating further from natural frequency. However, the response frequency of the downstream cylinder under constant Reynolds number is lower than that in varying Reynolds number conditions at high reduced velocity. The reduction in response amplitude and frequency can be attributed to the difference in Reynolds number at high reduced velocity in wakeinduced motions. Fig. 17 showed spectrum of lift forces on the cylinders and response amplitude of downstream cylinder at the same reduced velocity condition of Ur ¼ 20 but different Reynolds number conditions. Response behaviour of the two cases are very similar with single frequency amplitude response under multiple frequency force as consistently found in wake-induced motions. The Strouhal frequency of lift force component is higher with increasing of Reynolds number. It was reported in literature (Xu and Zhou, 2004) that there is strong dependency of Strouhal number on Reynolds number for cylinders in tandem at different tandem arrangement X=D (Xu and Zhou, 2004). classified relationship of St on Re into four categories based on the spacing distance between upstream and downstream cylinders. For X= D ¼ 3
5, there is a jump in St at Re ¼ 5  103
1:5  104 dependent on X=D. The St number continues increasing after the jump until it tapers to a constant value at Re ¼ 1:8  104 for all the spacing distances. It is believed that the jump and increase of St number may contribute to the difference in response amplitude and frequency of cylinders in comparison between constant and varying Re number wake-induced motions. Higher shedding frequency from upstream and downstream cylinder resulted in increasing of response amplitude as well as frequency as shown in Fig. 17. Fig. 18 further illustrated the effect of Reynolds number on forcing frequency and responses with increase of reduced velocity. It can be seen that the force frequency of upstream cylinder increases with increase of reduced velocity in varying Re number study while it remains the same for all reduced velocities in the case of constant Re number. This trend is consistently reflected in force spectrum of downstream cylinder including a shedding frequency and a lower frequency corresponding to its response frequency. The present numerical results have been consistent with earlier analysis (Assi et al., 2013) that WIV response amplitude is dependant on Reynolds number. The increase in amplitude at high reduced velocity is directly caused by the change in Reynolds number. In order to investigate further the effect of Re number on wake-induced motions, a constant Re number test was carried out at higher Reynolds number Re ¼ 3:05  104 to compare with (Hover and Triantafyllou, 2001) experiment and earlier study at Re ¼ 104 . The other conditions of the test are following the experimental setup where the mass ratio in this case is taken at m ¼ 3:0 slightly higher than the earlier varying Re number test. The distance between the two cylinders is adjusted to follow the experiment with X= D ¼ 4:75. Fig. 19 shows comparison of numerical prediction of response amplitude and frequency with experiment at the same Re number condition. Here the average 10% maximum amplitude (A=D1=10 ) is calculated from numerical simulation data and compared with the same data from earlier experiment (Hover and Triantafyllou, 2001). The data was also plotted against data from (Hover and Triantafyllou, 2001) for X=D ¼ 4:75. Numerical agrees well with experiment data at the same Re number condition for lower reduced velocity Ur  6 where vortex induced motions is dominated. At higher reduced velocity the average 1/10 amplitude is under-predicted in numerical simulations for the range of 8  Ur  17 tested in experiment. Results from present numerical study at higher reduced velocities (Ur > 17) showed that the amplitude is reaching to the maximum value of A=D1=10 ¼ 1:9 at Ur ¼ 22 and stays in that range until Ur ¼ 35. Though the current numerical results does not match well with experiment, it confirms a speculation from earlier experiment that large response amplitudes remains at higher reduced velocity. The response frequency is similarly found to be the same between numerical and experimental data. At high reduced velocity, the response frequency could be twice of the cylinder natural frequency as observed in both numerical simulations and experiment.

velocity conditions were normally altered to accommodate this variation. Due to changes in free stream velocity cylinders are effectively under varying Reynolds number conditions during those test conditions. It is believed that changes in Reynolds number together with reduced velocity variation will have coupled effects on responses of cylinders in wake-induced vibrations. The dependence of vortex induced vibrations on Reynolds number was experimentally studied in earlier work (Raghavan and Bernitsas, 2011) in which the authors showed a strong influence of Reynolds number of VIV response and characters of elastically mounted cylinder. As Reynolds number increases towards the end spectrum of subcritical region, shear layers are fully turbulent with high lift fluctuations thus inducing higher response amplitude even under large damping ratios. Recently effects of Reynolds number on wake-induced vibration was investigated in (Assi et al., 2013) to understand the concept of wake stiffness in response of cylinders in tandem. It was interestingly concluded that the response of cylinder in WIV region is directly influenced by Reynolds number and not reduced velocity. In an attempt to further understanding on the effects of Reynolds number on wake-induced vibrations, a numerical model of tandem cylinders was set-up following earlier experiment except that the Reynolds number was kept constant and the natural frequency of cylinder changed according to reduced velocity variation. In this work, numerical tests were performed at two fixed Reynolds number of 104 and 3  104 similar to the earlier experiment (Hover et al., 1997; Hover and Triantafyllou, 2001) in which a force feedback system was employed to control cylinder structural response while towing at a constant speed in a tank. These two Reynolds number regimes were also chosen as representatives of average and maximum Reynolds number conditions in (Assi et al., 2010b) experiment. Fig. 16 shows the response amplitude and frequency of cylinder under in-line tandem arrangement at X=D ¼ 5 with varying and constant Reynolds number. The numerical results at constant Reynolds number Re ¼ 104 was plotted against results from experiment for tandem cylinder at the varying Reynolds number conditions (Assi et al., 2010b; Chaplin and Batten, 2014) and single cylinder vortex induced vibration (Hover et al., 1997). There are clear discrepancies in results of response amplitudes and frequencies of downstream cylinder in wake-induced vibrations under constant and varying Reynolds number conditions. In the present constant Reynolds number tests, the amplitude response of downstream cylinder depicts similar trend as compared with single cylinder case at low reduced velocity Ur  6 obtained from earlier experiment (Hover et al., 1997). As shown in Fig. 16(a), the peak response of cylinder in tandem A=D  1:0 is observed at Ur ¼ 6 and very close to response of single cylinder in VIV lock-in region. This peak amplitude response of downstream cylinder at Re ¼ 104 is higher than cylinder in tandem with varying flow velocity test (Assi et al., 2010b) in VIV regime due to difference of Reynold number conditions. In earlier experiment (Assi et al., 2010b; Chaplin and Batten, 2014), Reynolds number is less than 5  103 for Ur  6 thus it is expected that response amplitude is lower. As it was shown earlier for single cylinder test at varying Reynolds number (Assi et al., 2010b), the peak response amplitude of A=D  0:8 was also lower than single cylinder study at Re ¼ 104 (Hover et al., 1997). The Reynolds number effect is consistently observed in the VIV regime of cylinder in tandem. Amplitude data for constant Reynolds number study is closer to experimental data for reduced velocity around Ur ¼ 10 where Reynolds number is around 104 in experimental study. At higher reduce velocities Ur > 10, response amplitude of downstream cylinder in the condition of constant Reynolds number deviates further from experimental data as reduced velocity increases. In contrast to increasing trend of response amplitude for varying Reynold number experiment, under constant Reynolds number condition the response amplitude is maintained at A=D  0:8 for a range of reduced velocity 10  Ur  20 before decreasing further with increases of reduced velocity. In Fig. 16(b), response frequency of downstream cylinder was plotted against reduced velocity. A similar trend in frequency is observed in both set-ups where frequency is increasing with reduced

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Acknowledgement

Fig. 20 showed response amplitude and phase angle of downstream cylinder at two different Reynolds number conditions of Re ¼ 104 and 3:05  104 . The response amplitude was computed as the standard rootmean-square values of motions and the Reynolds numbers were kept constant for varying of reduced velocities. The typical phase shift is observed in all the cases at reduced velocity corresponding to peak VIV response. The numerical results of phase shift agree well with experimental data. The response amplitudes show discrepancies between different Reynolds number conditions for a wide range of reduced velocities. However the amplitude tends to converge to a finite value of A= D  0:9 for Re ¼ 3:05  104 higher than A=D  0:5 for Re ¼ 104 as the reduced velocity increases. The current results is consistent with earlier similar constant Reynolds number test in (Assi et al., 2013) where response amplitudes asymptotically reach the values of free vibrating cylinders. Spectrum of response amplitude at different reduced velocities were plotted in Fig. 21 showing similar trend of increasing response frequencies with reduced velocity. The frequencies is slightly higher at higher Reynolds number conditions. It is clear that WIV response amplitude increases with increase of Reynolds number. Fig. 22 showed variation of root-mean-square values of lift force coefficient on the downstream cylinder with reduced velocities at Re ¼ 104 and Re ¼ 3:05  104 . In both cases, the peak value of force coefficients on cylinder in tandem was obtained in the VIV dominated region at low reduce velocity. As reduced velocity increases, lift force coefficient further reduces from the peak value and converges to a finite value of Cl;rms for downstream cylinder in tandem at different Reynolds number conditions. The constant lift coefficients are slightly lower in higher Reynolds number cases; thus inducing higher response amplitudes in wake-induced motions.

The authors would like to express gratitude to many fruitful discussions of the presented work with colleagues from SMMI-IHPC Joint Lab. This work is supported by the Agency for Science, Technology and Research (A*STAR) Science and Engineering Research Council grant SERC-132-183-0022. References Alam, M.M., Moriya, M., Takai, K., Sakamoto, H., 2003. Fluctuating fluid forces acting on two circular cylinders in a tandem arrangement at a subcritical Reynolds number. J. Wind Eng. Ind. Aerod. 91, 139–154. Armin, M., Srinil, N., 2013. Wake-induced transverse vibration of two interfering cylinders in tandem arrangement: modelling and analysis. In: 32nd International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers pp. V007T08A049–V007T08A049. Assi, G.R., 2014. Wake-induced vibration of tandem and staggered cylinders with two degrees of freedom. J. Fluid Struct. 50, 340–357. Assi, G., Bearman, P., Kitney, N., Tognarelli, M., 2010. Suppression of wake-induced vibration of tandem cylinders with free-to-rotate control plates. J. Fluid Struct. 26, 1045–1057. Assi, G., Bearman, P., Meneghini, J., 2010. On the wake-induced vibration of tandem circular cylinders: the vortex interaction excitation mechanism. J. Fluid Mech. 661, 365–401. Assi, G., Bearman, P., Carmo, B., Meneghini, J., Sherwin, S., Willden, R., 2013. The role of wake stiffness on the wake-induced vibration of the downstream cylinder of a tandem pair. J. Fluid Mech. 718, 210–245. Bokaian, A., Geoola, F., 1984. Wake-induced galloping of two interfering circular cylinders. J. Fluid Mech. 146, 383–415. Borazjani, I., Sotiropoulos, F., 2009. Vortex-induced vibrations of two cylinders in tandem arrangement in the proximity–wake interference region. J. Fluid Mech. 621, 321–364. Brika, D., Laneville, A., 1999. The flow interaction between a stationary cylinder and a downstream flexible cylinder. J. Fluid Struct. 13, 579–606. Chaplin, J., Batten, W., 2014. Simultaneous wake-and vortex-induced vibrations of a cylinder with two degrees of freedom in each direction. J. Offshore Mech. Arctic Eng. 136, 031101. Dong, S., Karniadakis, G.E., Ekmekcia, A., Rockwell, D., 2006. A combined direct numerical simulation and particle image velocimetry study of the turbulent near wake. J. Fluid Mech. 569, 185–207. Facchinetti, M., de Langre, E., Fontaine, E., Bonnet, P., Etienne, S., Biolley, F., 2002. Viv of two cylinders in tandem arrangement: analytical and numerical modeling. In: Proceedings of the 12th International Offshore and Polar Engineering Conference, pp. 524–531. Feymark, A., Alin, N., Bensow, R., Fureby, C., 2012. Numerical simulation of an oscillating cylinder using large eddy simulation and implicit large eddy simulation. J. Fluid Eng. 134, 031205. Gallardo, D., Bevilacqua, R., Sahni, O., 2014. Data-based hybrid reduced order modeling for vortex-induced nonlinear fluidstructure interaction at low Reynolds numbers. J. Fluid Struct. 44, 115–128. Hover, F.S., Triantafyllou, M.S., 2001. Galloping response of a cylinder with upstream wake interference. J. Fluid Struct. 15, 503–512. Hover, F., Miller, S., Triantafyllou, M., 1997. Vortex-induced vibration of marine cables: experiments using force feedback. J. Fluid Struct. 11, 307–326. Huera-Huarte, F., Bearman, P., 2011. Vortex and wake-induced vibrations of a tandem arrangement of two flexible circular cylinders with near wake interference. J. Fluid Struct. 27, 193–211. Igarashi, T., 1981. Characteristics of the flow around two circular cylinders arranged in tandem (1st report). Bull. JSME 24, 323–331. Kitagawa, T., Ohta, H., 2008. Numerical investigation on flow around circular cylinders in tandem arrangement at a subcritical Reynolds number. J. Fluid Struct. 24, 680–699. Ljungkrona, L., Norberg, C., Sunden, B., 1991. Free-stream turbulence and tube spacing effects on surface pressure fluctuations for two tubes in an in-line arrangement. J. Fluid Struct. 5, 701–727. Menter, F., Kuntz, M., 2004. Adaptation of eddy-viscosity turbulence models to un- steady separated flow behind vehicles. In: The Aerodynamics of Heavy Vehicles: Trucks, Buses, and Trains, p. 567. Mittal, S., Kumar, V., 2001. Flow-induced oscillations of two cylinders in tandem and staggered arrangements. J. Fluid Struct. 15, 717–736. Nguyen, V.-T., Nguyen, H.H., 2016. Detached eddy simulations of flow induced vibrations of circular cylinders at high Reynolds numbers. J. Fluid Struct. 63, 103–119. Nguyen, V.-T., Nguyen, H.H., Hans, H., Tutty, O., Weymouth, G., 2015. Detached Eddy Simulations of Wake Induced Vibrations for Cylinders in Tandems at High Reynolds Numbers. URL: https://www.onepetro.org/conference-paper/ISOPE-I-15-702. Norberg, C., 2003. Fluctuating lift on a circular cylinder: review and new measurements. J. Fluid Struct. 17, 57–96. Paidoussis, M., Price, S., 1988. The mechanisms underlying flow-induced instabilities of cylinder arrays in crossflow. J. Fluid Mech. 187, 45–59. Raghavan, K., Bernitsas, M., 2011. Experimental investigation of Reynolds number effect on vortex induced vibration of rigid circular cylinder on elastic supports. Ocean Eng. 38, 719–731.

4. Conclusions This work had presented the application of numerical simulation approach for investigation of wake-induced vibrations in high Reynolds number flows. The hybrid RANS/LES approach has been successful in predicting wake-induced motions of cylinders in tandem at spacing distance X=D ¼ 4
5 and Reynolds number Re ¼ 5  103
3  104 . We validated the numerical results against two representative experiment with reasonably good agreement. Distinct WIV responses of large amplitude outside of typical VIV lock-in regions was well captured in numerical model. It was also able to predict increase of response frequency with velocity increases as reported in various experimental study. Complex interactions between cylinders wakes and downstream cylinder motions resulted in finite cross flow forces acting on the cylinder. The numerical results further illustrate the analysis on the mechanism of WIV where the vibration of downstream cylinder is cased by restoring lift fluctuation resulted from its interaction with upstream cylinder wake. The effect of Reynolds number in wake-induced vibrations were analysed using the proposed numerical approach. It confirmed that Reynolds number has a strong influence on wake-induced motions. Higher Reynolds number caused larger wake-induced vibration amplitude. Responses of tandem cylinders in varying reduced velocity at constant Reynolds number revealed a different characteristic of response amplitude at high reduced velocity. While the WIV amplitude was shown to remain large at high reduced velocities, it tends to gradually taper off with increase of reduced velocity to a finite amplitude considered as free WIV amplitude. This free vibration amplitude decreases as Reynolds number reduces. The current study has illustrated great potential of numerical simulation tools for investigation of wake-induced vibration through extensive study of the phenomena. However there remains many outstanding issues to be addressed in order to fully understanding the wake-induced vibrations including the key mechanism governing the response frequency of a cylinder in tandem. It will form the main scope of our future work in this direction.

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Ocean Engineering 154 (2018) 341–356 Travin, A., Shur, M., Strelets, M., Spalart, P., 2000. Detached-eddy simulations past a circular cylinder. Flow, Turbul. Combust. 63, 293–313. Williamson, C., Govardhan, R., 2004. Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36, 413–455. Xu, G., Zhou, Y., 2004. Strouhal numbers in the wake of two inline cylinders. Exp. Fluid 37, 248–256. Zdravkovich, M., 1977. Review of flow interference between two circular cylinders in various arrangements. J. Fluid Eng. 99, 618–633.

Rosetti, G.F., Vaz, G., Fujarra, A.L.C., 2012. URANS calculations for smooth circular cylinderflow in a wide range of Reynolds numbers: solution verification and validation. J. Fluid Eng. 134, 121103–121118. Spalart, P.R., 2009. Detached-eddy simulation. Annu. Rev. Fluid Mech. 41, 181–202. Spalart, P., Deck, S., Shur, M., 2006. A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theor. Comput. Fluid Dynam. 20, 181–195. Strelets, M., 2001. Detached Eddy Simulation of Massively Separated Flows. AIAA–2001–0879. Sumner, D., 2010. Two circular cylinders in cross-flow: a review. J. Fluid Struct. 26, 849–899.

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