Wake-induced vibrations of an elastically mounted cylinder located downstream of a stationary larger cylinder at low Reynolds numbers

Journal of Fluids and Structures 50 (2014) 479–496

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Wake-induced vibrations of an elastically mounted cylinder located downstream of a stationary larger cylinder at low Reynolds numbers Huakun Wang a, Wenyu Yang a, Kim Dan Nguyen b, Guoliang Yu a,n a State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, No. 800, Dongchuan Road, Shanghai 200240, China b Laboratory for Hydraulics Saint-Venant, Université Paris-Est, 6 quai Watier, 78400 Chatou, France

a r t i c l e i n f o

abstract

Article history: Received 1 August 2013 Accepted 7 July 2014 Available online 19 August 2014

Two-dimensional numerical simulations of flow past two unequal-sized circular cylinders in tandem arrangement are performed at low Reynolds numbers (Re). The upstream larger cylinder is stationary, while the downstream cylinder has both one (transverseonly) and two (transverse and in-line) degrees of freedom (1-dof and 2-dof, respectively). The Re, based on the free stream velocity U1 and the downstream cylinder diameter d, varies between 50 and 200 with a wide range of reduced velocities Ur. The diameter of the upstream cylinder is twice that of the downstream cylinder, and the center-to-center spacing is 5.5d. In general, for the 1-dof case, the calculations show that the wake-induced vibrations (WIV) of the downstream cylinder are greatly amplified when compared to the case of a single cylinder or two equal-sized cylinders. The transverse amplitudes build up to a significantly higher level within and beyond the lock-in region, and the Ur associated with the peak amplitude shifts toward a higher value. The dominant wake pattern is 2S mode for Re¼ 50 and 100, while with the increase of Re to 150 and 200, the P þ S mode can be clearly observed at some lower Ur. For the 2-dof vibrations, the transverse response characteristics are similar to those presented in the corresponding 1-dof case. The in-line responses are generally much smaller, except for several significant vibrations resulting from in-line resonance. The obvious in-line vibration may induce a C (chaotic) vortex shedding mode for higher Re (Re ¼200). With regard to the 2-dof motion trajectories, besides the typical figure-eight pattern, several odd patterns such as figure-double eight and single-looped trajectories are also obtained due to the wake interference effect. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Two unequal-sized circular cylinders Tandem arrangement Wake-induced vibrations Low Reynolds number

1. Introduction Vortex-induced vibration (VIV) of cylindrical structures is a subject that has attracted a lot of attentions due to its importance in engineering applications and in fluid dynamics research. Extensive studies have been mainly focused on 1-dof VIV of an elastically mounted circular cylinder in the transverse flow direction. It is well known that the lock-in or synchronization phenomenon occurs when the vibration frequency approaches the structural natural frequency. Different

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Corresponding author. Tel.: þ86 13482150851. E-mail address: [email protected] (G. Yu).

http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.006 0889-9746/& 2014 Elsevier Ltd. All rights reserved.

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vortex shedding patterns, such as 2S mode (two single vortices are shed per cycle), 2P mode (two pairs of vortices are released per cycle) and P þS mode (a single vortex and a pair of vortices are released per cycle) can be observed depending on some important parameters such as the Reynolds number, the reduced velocity and the cylinder mass. The 2-dof VIV of a circular cylinder has also been studied. An interesting feature is that the motion orbit usually presents a figure-eight pattern because the frequency of the drag force is approximately twice that of the lift force. More details about the VIV of a single circular cylinder can be found in review papers by Williamson and Govardhan (2004), Sarpkaya (2004), Gabbai and Benaroya (2005), and Bearman (2011). Multiple cylinders are more often encountered in practical engineering. When an elastically mounted cylinder is immersed in the wake of another bluff body, the wake-induced vibration (WIV) of the downstream cylinder is usually completely different from the VIV of a single cylinder in free stream. Many previous investigations, experimentally and numerically, have been focused on the WIV response involving two equal-sized circular cylinders (Assi et al., 2006, 2010; Bao et al., 2011; Bokaian and Geoola, 1984; Brika and Laneville, 1999; Carmo et al., 2007, 2011; Hover and Triantafyllou, 2001; King and Johns, 1976; Zdravkovich, 1985). As confirmed in these studies, the wake interference was able to lead to a wider lock-in range, and within this range the transverse vibration amplitudes of the downstream cylinder were larger than those of a single cylinder. Some experimental observations at high Re have also shown that the cylinders subjected to wake interference could experience galloping response (Assi et al., 2010; Bokaian and Geoola, 1984; Brika and Laneville, 1999; Hover and Triantafyllou, 2001). In addition, using two- and three-dimensional (2D and 3D) numerical simulations, Carmo et al. (2011) studied the WIV of two equal-sized cylinders in tandem arrangement at Re ¼150 and 300. The upstream cylinder was fixed and the downstream cylinder was free to vibrate only transverse to the flow. Their results showed that the downstream cylinder sustained high-amplitude vibrations for higher reduced velocities outside the lock-in range, but not wake-induced galloping. The question would be: is there an upper limit of reduced velocity at which these vibrations cease? Carmo et al. (2013) conducted a further numerical study at infinite reduced velocity for 100rRer645. They confirmed that there was not a delimited range of reduced velocity, within which the vibration occurred with significant amplitude. In contrast, up to now there only exist a few studies on the WIV involving two unequal-sized cylinders. Lam and To (2003) experimentally investigated the transverse vibration of a flexibly mounted cylinder placed downstream of a larger cylinder at 720rRer3600. The interfering larger cylinder had a diameter twice that of the vibrating cylinder. The authors observed that for spacing up to 2.34 times the larger cylinder diameter in the tandem arrangement, the flexible cylinder still exhibited negligible vibrations at nearly all reduced velocities, due to the shielding of the larger cylinder. Later on, To and Lam (2007) carried out an experimental study for the flexibly mounted cylinder placed upstream of the larger cylinder. They reported that galloping-type vibrations occurred when the flexible cylinder was placed just in front of the larger cylinder. Recently, Gao et al. (2014) experimentally studied the transverse vibration of a smaller cylinder in the wake of a larger cylinder for tandem arrangement at Re ¼7200. The diameter ratio between two cylinders was 0.75. Their results revealed that the vibration amplitude of the smaller cylinder was lower than that of a single cylinder under the same flow condition, even though the spacing between two cylinders became sufficiently large. However, little information about the lock-in response was given because the reduced velocity in their computations was fixed at a value of 7. At the present state of the art, the studies on the WIV characteristics of two unequal-sized cylinders, such as the lock-in responses, the flow patterns and the motion trajectories, are still lacking. The main purpose of this paper is to fill the knowledge gap on both 1-dof and 2-dof WIV of an elastically mounted cylinder behind a stationary larger cylinder and supplement the findings reported by others, using 2D numerical simulation. The diameter of the upstream cylinder is twice that of the vibrating cylinder. The two cylinders are arranged in tandem with a center-to-center spacing of 5.5d. This spacing is expected to be sufficient to form strong wake interference between two cylinders. The Reynolds number is limited in the range of 50rRer200, which has been widely considered in previous numerical studies. For each fixed Re, the reduced velocity ranges from 3.0 to 30.0 by changing the structural natural frequency. In this work, we try to answer the following questions: (i) Could the lock-in phenomenon occur? (ii) Could the downstream cylinder exhibit large-amplitude transverse vibration associated with the lock-in? (iii) How different would be the WIV responses of two unequal-sized cylinders, compared to those for the case of a single cylinder or two equal-sized cylinders? (iv) What kinds of flow patterns would appear? (v) Whether the in-line freedom has any effects on the transverse response? (vi) Which kinds of trajectories would the downstream cylinder with 2-dof move along? The outline of this paper is organized as follows. Section 2 describes the mathematical and numerical background, including the numerical techniques to solve the fluid flow, the cylinder motion, the mesh motion and the fluid–solid interaction (FSI). Section 3 presents the physical description of the studied problem as well as the validation of the used computational code. The results and discussions are given in Section 4 and Section 5. Section 6 attempts to draw conclusions for this study and to deliver the answers to the above-mentioned questions. 2. Mathematic and numerical background 2.1. Equations for the incompressible fluid flow The governing equations for fluid flow are the two-dimensional incompressible Navier–Stokes equations. The arbitrary Lagrangian–Eulerian (ALE) scheme (Donea et al., 1982) is applied to deal with the flow problem with moving boundaries.

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In the ALE description, the Navier–Stokes equations in tensor forms are written as follows: ∂ui ∂u ∂σ ij þ cj i ¼ ; ∂t ∂xj ∂xj

ð1Þ

∂ui ¼ 0; ∂xi

ð2Þ

where ui is the velocity component in the xi-direction (i ¼1, 2 or x, y); t is the time; cj is the jth component of fluid velocity relative to moving mesh, defined as cj ¼uj  wj, where wj is the mesh velocity. The stress tensor σij is expressed as
 1 ∂ui ∂uj σ ij ¼ pδij þ þ ; ð3Þ Re ∂xj ∂xi wherein p is the pressure, δij is the identity tensor, and Re is the Reynolds number defined as Re ¼U1 d/ν, where ν is the kinematic viscosity of the fluid. The Navier–Stokes equations are supplemented by the Dirichlet and Neumann-type boundary conditions:  ui jΓg ¼ g i ; σ ij nj Γ ¼ hi ; ð4Þ h

where Гg and Гh are two nonoverlapping subsets of the piecewise smooth domain boundary; gi and hi denote the values prescribed on Гg and Гh, respectively; and nj is the jth component of the outward unit vector normal to Гh. The time integration of the momentum equation (Eq. (1)) is performed by a four-step fractional method (Choi et al., 1997; Baek and Sung, 1998). In this scheme, the intermediate velocities are first calculated by decoupling the pressure gradient term from Eq. (1). However, the velocity field does not satisfy the continuity equation (Eq. (2)). To amend it, the pressure is obtained by imposing the continuity constraint, and the velocity is finally corrected by the new pressure field. The procedure of the semi-implicit four-step fractional method is presented as follows: " !
 # n ^ u^ i uni ∂uni ∂uj 1 ∂u^ i ∂u^ j ∂pn n ∂ui =∂xj þ ∂ ∂ þ cj ¼ þ þ ; ð5Þ =xj  Δt ∂xj 2Re ∂xj ∂xi ∂xj ∂xi ∂xi uni  u^ i ∂pn ¼ ; Δt ∂xi

ð6Þ

∂2 pn þ 1 1 ∂uni ¼ ; Δt ∂xi ∂xj ∂xj

ð7Þ

uni þ 1  uni ∂pn þ 1 ¼ ; Δt ∂xi

ð8Þ

where u^ i and uni are the intermediate velocities, and Δt is the time increment. The superscripts n and nþ1 are the time points of tn and tn þ 1 (Δt¼tn þ 1  tn). The linearized convection term is used to avoid nonlinear iteration, and the diffusion term is integrated using the second-order Crank–Nicolson method. Eqs. (5)–(8) are solved by a Galerkin finite element method (FEM). For the convection dominated flow, in order to eliminate the numerical oscillation in the velocity solutions, a streamline upwind/Petrov–Galerkin (SUPG) scheme is applied to the spatial discretization of Eq. (5). In the SUPG scheme, an artificial diffusion term is added to Eq. (5) with an appropriate weighting function chosen as τs cnk ∂δui =∂xk ; where δui is the velocity shape function, and the parameter τs is defined as (Choi et al., 1997) e

τs ¼

zh ; 2jjce jj

z ¼ cothðPeÞ 

1 ; Pe

e

Pe ¼

jjce jjh ; 2ν

ð9Þ

where he, ce, and Pe represent the characteristic element length, the convection velocity at the element center, and the element Peclet number, respectively. The velocity and the pressure are interpolated by the linear shape function: ui ¼ uiI ΦI ;

p ¼ pI ΦI ;

ð10Þ

where uiI and pI are the values of ui and p at node I, and ΦI is the shape function of the element. Multiplying Eqs. (5)–(8) by shape function and integrating over the computational domain, and adding the SUPG term, the matrix equations of the four steps can be obtained as h  i ð11Þ M nsIJ þ Δt C nIJ þ 12 DIJ u^ iJ ¼ MnsIJ uniJ  12 ΔtDIJ uniJ þ ΔtðGniIJ pnJ þ H iI Þ; M IJ uniJ ¼ M IJ u^ iJ þ ΔtAiIJ pnJ ; ðBIJ EIJ ÞpnJ þ 1 ¼ 

1 A un ; Δt iIJ iJ

ð12Þ ð13Þ

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M IJ uniJ þ 1 ¼ M IJ uniJ  ΔtAiIJ pnJ þ 1 ;

ð14Þ

where

Z Z ∂ΦI M nsIJ ¼ ΦI ΦJ dΩ þ τs cnk ΦJ dΩ; ∂xk Ω Ω Z Z ∂ΦJ ∂ΦI n ∂ΦJ dΩ þ τs cnk c dΩ; C nIJ ¼ ΦI cnj ∂x ∂xk j ∂xj j Ω Ω Z Z Z 1 ∂ΦI ∂ΦJ ∂ΦI ∂ΦI ∂ΦJ DIJ ¼ dΩ; GniIJ ¼ ΦJ dΩ τs cnk dΩ; ∂xk ∂xi Ω Re ∂xj ∂xj Ω ∂xi Ω Z Z Z ∂ΦJ H iI ¼ ΦI hi dΓ; MIJ ¼ ΦI ΦJ dΩ; AiIJ ¼ ΦI dΩ; ∂xi Γ Ω Ω Z Z ∂ΦJ ∂ΦI ∂ΦJ dΩ; EIJ ¼ ΦI n dΓ: BIJ ¼ ∂xi i Ω ∂xi ∂xi Γ

ð15Þ

2.2. Equations for the cylinder motion The dynamic behavior of the downstream vibrating cylinder with 2-dof is modeled via a mass-spring-damper system as follows (Prasanth and Mittal, 2009): ∂2 X 4πζ ∂X 4π 2 2C D þ þ X¼ ; πM r ∂t 2 U r ∂t U 2r

ð16Þ

∂2 Y 4πζ ∂Y 4π 2 2C L þ þ Y¼ ; πM r ∂t 2 U r ∂t U 2r

ð17Þ

where X and Y denote the in-line and transverse displacements, respectively. ζ is the structural damping ratio, and Mr is the mass ratio, expressed as Mr ¼4m/(πρd2), where m is the mass of the cylinder per unit span. The reduced velocity, Ur, is defined as Ur ¼U1/(fnd), where fn is the structural natural frequency. CD and CL are the drag and lift coefficients, respectively, computed by R 2 Γcyl σ 1j nj dΓ ; ð18Þ CD ¼ ρU 21 d CL ¼

R 2 Γcyl σ 2j nj dΓ ρU 21 d

;

ð19Þ

where nj is the jth component of the unit vector normal to the cylinder boundary Гcyl. The temporal integrations of the dynamic equations are numerically performed by using Newmark's (1959) algorithm. 2.3. Equations for the mesh motion In order to accommodate the cylinder motion, a mesh-updating scheme is incorporated into the fluid–solid coupling algorithm. The calculation of the mesh node displacements is based on the boundary-value problem of a modified Laplace equation (Masud et al., 2007): ∂2 ½ð1 þ γÞSj  ¼ ∂xi ∂xi

 Sj 

Γm

0;

¼ gj ;

 Sj Γ ¼ 0; f

ð20Þ

where Sj denotes the displacement of the mesh node in the xj-direction; Γm and Γf are the moving and fixed portions of the boundary respectively; gj is the jth component of the mesh displacement prescribed on Γm; and γ is a parameter that controls the mesh deformation. In order to avoid excessive deformation of the near-wall elements, the parameter γ in an element is set to be γe ¼(1 Δmin/Δmax)/(Δe/Δmax), where Δe, Δmin, and Δmax denote the areas of the current, the largest and smallest elements in the mesh, respectively. By specifying the displacements at all the boundaries, Eq. (20) is solved by a Galerkin FEM. 2.4. Fluid–solid interaction procedure The resolution of the fluid–solid interaction (FSI) problem can be conducted without difficulty by using a staggered algorithm (Placzek et al., 2009), in which the flow field and the structure motion are solved successively for a given time step. This algorithm is described as follows: (i) Initialize the flow field on the initial mesh; (ii) Calculate the hydrodynamic

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force coefficients (CD and CL) of the cylinder by Eqs. (18) and (19); (iii) Compute Eqs. (16) and (17) to get the displacement and velocity of the cylinder; (iv) Update the mesh field based on the modified Laplace equation (Eq. (20)); (v) Solve Eqs. (5)–(8) to obtain the fluid velocity and pressure on the new mesh; (vi) Go to step (ii) and continue the loop until the time limit of the computation is reached. Compared with the strong coupling strategy, the staggered algorithm may lead to the divergence of the FSI system for very low mass ratios. However, many previous numerical studies (Bao et al., 2011, 2012; Borazjani and Sotiropoulos, 2009; Han et al., 2014; He et al., 2012; Placzek et al., 2009) have demonstrated that for the range of mass ratio considered in our work (Mr ¼ 2.546), the staggered algorithm can yield robust FSI iterations and obtain satisfactory results.

3. Problem description WIV simulations are performed for two unequal-sized cylinders in tandem arrangement. The upstream larger cylinder is stationary, whereas the downstream smaller cylinder is free to vibrate with both 1-dof and 2-dof. The diameter ratio of 2.0 between two cylinders is selected as a representative case to explore the response characteristics of two unequal-sized cylinders. This diameter ratio has also been considered in previous study by Lam and To (2003), who carried out a series of experiments on WIV of two unequal-sized cylinders. The Re in our work ranges from 50 to 200 based on the free stream velocity U1 and the downstream cylinder diameter d, enabling a 2D numerical investigation. The center-to-center spacing between two cylinders is set as 5.5d. The mass ratio of the downstream cylinder is Mr ¼2.546 and the structural damping coefficient ζ is neglected in order to excite large-amplitude vibration. For each fixed Re, the reduced velocity Ur is systematically increased in the range of 3.0–30.0 by changing the structural natural frequency. Additionally, for the 1-dof case, the vibration responses of the downstream cylinder at infinite Ur (with no spring and damper attached to it) are also studied.

3.1. Computational domain and boundary conditions The geometric configuration and boundary conditions are shown in Fig. 1. As stated by Prasanth and Mittal (2008), the blockage has a significant effect on the flow past bluff bodies, and a blockage of more than 2.5% can lead to incorrect prediction of hydrodynamic forces and vibration responses. Therefore, in our simulations, the computational domain is defined as Ω¼[ 50d, 100d]  [ 50d, 50d], resulting in a blockage of 2%. Dirichlet conditions are imposed upon the inlet boundary as u1 ¼ U 1 and u2 ¼ 0. At the outlet, Neumann conditions are applied as ∂u1 =∂x ¼ 0 and ∂u2 =∂x ¼ 0, and the pressure is set to 0. The lateral boundaries have slip conditions of ∂u1 =∂y ¼ 0 and u2 ¼ 0, while no-slip conditions are applied on the surfaces of the cylinders. All WIV simulations are initialized from the solution of the stationary counterpart.

3.2. Mesh resolution study In order to ensure the mesh independency, computations were performed with three different meshes (a coarse mesh M1, a medium mesh M2, and a fine mesh M3) for the case of Re¼100 with Ur ¼11.0. The time step corresponding to the mesh resolution is set as Δt ¼0.002. The mesh parameters, the vibration amplitudes, and the statistics of hydrodynamic forces for the downstream cylinder with 2-dof are given in Table 1, along with the percentage changes. As shown in the table, the values obtained from the three meshes are not sensitive to the mesh size. The maximum change of 2.73% occurs in the value of C L' from M1 to M2, while the maximum difference is reduced to only 1.06% on M3. This test establishes the adequacy of M2 in the present simulations. Therefore, all of the results in this study are computed using M2. The whole finite element mesh of M2 which consists of 113 491 elements and 57 196 nodes is shown in Fig. 2(a), with a close-up view in Fig. 2(b).

Fig. 1. WIV of an elastically mounted cylinder behind a stationary larger cylinder: schematic of the computational domain and boundary conditions.

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Table 1 The values of global parameters for the downstream cylinder with 2-dof (Re ¼ 100, Ur ¼ 11.0). Mesh

Nodes

Elements

Xmax/d

Ymax/d

CD

C 0L

M1 M2 M3

48057 57196 67254

95333 113491 133487

0.0607 0.0611 (0.66%) 0.0612 (0.16%)

1.604 1.617(0.81%) 1.621(0.24%)

1.235 1.251 (1.30%) 1.257 (0.48%)

0.183 0.188 (2.73%) 0.190 (1.06%)

Fig. 2. WIV of two unequal-sized cylinders in tandem: (a) the entire computational mesh and (b) close-up view of the mesh near the cylinders.

Fig. 3. Comparisons of the maximum amplitude responses for a single cylinder subjected to transverse-only vibration (Re ¼150).

3.3. Code validation test The FSI code is validated by applying it to the VIV of a single cylinder with 1-dof (transverse-only). Fig. 3 shows the response curve of the maximum vibration amplitude versus Ur at Re¼150 with Mr ¼2.546 and ζ¼0, as well as the available data in the literatures (Ahn and Kallinderis, 2006; Bao et al., 2012; Borazjani and Sotiropoulos, 2009). A good agreement between our results and previous studies is obtained. The maximum amplitudes for all of the response curves occur at the same reduced velocity of Ur ¼4.0. The resonant regions cover a range of 4.0r Ur r7.0. Outside this region, the maximum amplitudes suddenly decrease to a very low value. The validation test proves the reliability of our FSI code in accurately calculating the VIV response of a single cylinder. Its application with confidence to further WIV studies on two cylinders is approved. 4. Results of WIV with 1-dof 4.1. Frequency analysis It is interesting to explore how a stationary larger cylinder affects the frequency characteristics of a downstream freely vibrating cylinder. Fig. 4 shows the frequency analysis of 1-dof WIV of two unequal-sized cylinders for different Re. The foy denotes the transverse vibration frequency of the downstream cylinder, St is the Strouhal number for the corresponding Re,

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Fig. 4. Frequency analyses of WIV of two unequal-sized cylinders for different Re (1-dof): (a) Re¼50; (b) Re¼100; (c) Re¼ 150; and (d) Re¼ 200.

and fv represents the vortex shedding frequency of the upstream cylinder. It is evident that the vibration frequency of the downstream cylinder seriously deviates from the Strouhal line, but was completely dominated by the vortex shedding of the upstream cylinder. Due to the wake interference of the upstream cylinder twice as large, the vibration frequencies of the downstream one are reduced to nearly half of the Strouhal number. As mentioned in previous studies, when the vibration frequency approaches the structural natural frequency, the vortex-excited body can undergo a resonance phenomenon, known as ‘lock-in’, characterized by wide frequency range and large amplitude. It is worth noting that the vibration frequency needs not to be equal to the structural natural frequency, and there is usually a slight detuning, i.e. ‘soft’ lock-in (Borazjani and Sotiropoulos, 2009; Mittal and Kumar, 2001). Mittal and Kumar (2001) pointed out that ‘soft’ lock-in is one of the mechanisms of the nonlinear oscillator to self-limit its vibration amplitude. In the current simulations, as shown in Fig. 4(a), the lock-in of the downstream cylinder covers a wide range of 11rUr r17 for Re¼50; as the Re varies from 100 to 200 (Fig. 4(b)–(d)), the lock-in occurs within a relatively narrow band of reduced velocity around Ur ¼9, but it is difficult to define the lower and upper boundaries. The large-amplitude vibrations associated with the lock-in will be discussed in the next section. 4.2. Transverse vibration amplitudes Fig. 5(a) shows the maximum transverse amplitude of the downstream cylinder versus Ur for different Re. For Re¼50, the transverse response appears to be negligible at Ur r10, but shows a sudden jump at Ur ¼ 11 which is related to the onset of lock-in, then it reaches the peak value (Ymax/d¼ 1.385) at the resonance reduced velocity Ur ¼13, and decreases slowly afterwards. For Re¼100, 150 and 200, the vibration responses are very similar: the transverse amplitudes increase smoothly and obtain the peak values (1.513 rYmax/dr1.557) at Ur ¼11 in the lock-in region, then begin to drop. Assi et al. (2010) pointed out that the WIV of the downstream cylinder was excited by the unsteady vortex-structure interactions between the body and the upstream wake. A typical feature of WIV is the large amplitude persisting to high reduced velocities (Assi et al., 2010; Bao et al., 2011; Carmo et al., 2011; Hover and Triantafyllou, 2001). This response characteristic is reproduced in the current simulations. Fig. 6 illustrates the flow fields at selected instants during one oscillation circle for a representative case of (Re, Ur)¼(50, 18). As can be seen, the passage of flow through the gap region triggers a sequence of

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Fig. 5. Variations of the maximum transverse amplitude for the downstream cylinder versus (1-dof): (a) Ur for different Re and (b) Re at infinite Ur.

Fig. 6. Times histories of lift coefficient and displacement of an elastically mounted cylinder in the wake of a stationary larger cylinder at Ur ¼ 18 for Re¼50 (left column). Pressure contours at selected time points (i), (ii) and (iii) (middle column). Vorticity contours and streamlines at selected time points (i), (ii) and (iii) (right column).

vortex–cylinder interactions that dominate the flow dynamics. The vortex impingement point and the associated lowpressure core shift periodically along the wall of the downstream cylinder, inducing lift fluctuations which are not synchronized with the cylinder motion. The favorable phase lag between the lift force and the displacement guarantees the energy transfer from the fluid to the structure and sustains the severe vibrations (Assi et al., 2010; Bao et al., 2011). The low-amplitude vibrations at Ur r10 for Re¼50 can be ascribed to the shielding of the upstream larger cylinder. Fig. 7 shows the flow fields for a case of (Re, Ur)¼(50, 10). It can be observed that the flow separated from the upstream larger cylinder always passes though the top and bottom sides of the downstream cylinder, and highly stretches in the flow direction; meanwhile, two symmetric recirculation zones develop in the gap between the cylinders, inducing a nearly stagnant symmetric low-pressure pocket about the wake centreline. Consequently, the balance of this pressure distribution leads to nearly zero lift force and slight vibration of the downstream cylinder. Lam and To (2003) conducted experiments in a wind tunnel (Re up to 3600) with a flexibly mounted cylinder placed in the vicinity of a larger cylinder and subjected to cross-flow (D/d ¼2; 0 rX0/Dr2.34 and 0 rY0/Dr1.41; X0 and Y0 denote the horizontal and vertical center-to-center distances, respectively). They proposed that the modifications of the vibration responses were related to the presence or

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Fig. 7. Times histories of lift coefficient and displacement of an elastically mounted cylinder in the wake of a stationary larger cylinder at Ur ¼10 for Re ¼50 (left column). Pressure contours at selected time points (i), (ii) and (iii) (middle column). Vorticity contours and streamlines at selected time points (i), (ii) and (iii) (right column).

absence of gap flow between the cylinders. For the tandem arrangement in their experiment, the vibrations of the flexible cylinder were negligibly small at almost all reduced velocities since the gap flow between two cylinders was always suppressed. This phenomenon is similar to those observed in our simulations for Ur r10 and Re¼50. However, in other cases tested here, the contribution of flow through the gap region can be considerable for the excitation of the WIV response. Compared to the case of a single cylinder or two equal-sized cylinders, the WIV response of two unequal-sized cylinders has its unique characteristics. Take the case of Re ¼150 as an example, the transverse peak amplitude of the downstream cylinder in the wake of a larger cylinder is observed at Ur ¼11.0 and its value is as high as 1.557d (see Fig. 5), approaching nearly three times that of the single cylinder observed at Ur ¼4.0 (see Fig. 3); in contrast, Carmo et al. (2011) reported that the vibration amplitude of the cylinder behind a equal-sized cylinder reached a peak value of 0.9D at Ur ¼6.0 (D was the cylinder diameter; center-to-center spacing Lx/D varied from 1.5 to 8; Re¼150). This can be explained by the fact that the presence of the upstream larger cylinder intensifies the gap flow along with the vortex–cylinder interactions, which is responsible for the amplified vibration of the downstream cylinder; in addition, the larger cylinder slows down the vortex shedding frequency in the wake, giving rise to a shift in the peak response of the rear cylinder to a higher Ur. Fig. 5(b) shows the variations of the transverse response of the downstream cylinder against Re at infinite Ur. To obtain a higher resolution, supplementary simulations have been performed within 50rRer200. It can be seen from the figure that, the high-amplitude vibrations still persist, and decrease monotonically with the increase of Re in the whole range of 50rRer200. For the 2D WIV simulation of two equal-sized cylinders, a similar observation was also mentioned by Carmo et al. (2013), but differently, they reported the upper Re limit for the monotone reduction region of the vibration amplitude was Re¼165, which approached the critical Re regarding 3D instabilities of the flow. A possible explanation for this difference is that, in our study, the 3D effects may be weakened due to the higher-amplitude vibration of the downstream cylinder, so this upper Re limit is increased. On the other hand, Borazjani and Sotiropoulos (2009) proposed that even though the flow did transition to a weakly 3D state at Re¼200, the 3D structures were not intense enough to affect the dynamic response of the vibrating system, which was essentially identical to that got from the 2D simulations.

4.3. Flow patterns The flow past multiple bluff bodies becomes very complex when one or more bodies are free to vibrate (Prasanth and Mittal, 2009). In the current simulations, since the the elastically mounted cylinder is located in the wake of the stationary larger one, flow interference occurs. Fig. 8 summarizes the near wake patterns behind the downstream cylinder. It is found

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Fig. 8. Flow patterns behind two unequal-sized cylinders for different Re and Ur (1-dof).

Fig. 9. Instantaneous vorticity fields for the fully developed flow past a stationary larger cylinder and an elastically mounted cylinder at selected Re and Ur (1-dof).

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Fig. 10. Variations of the drag and lift coefficients of the downstream cylinder versus Ur for different Re (1-dof): (a) mean drag coefficient; (b) r.m.s. drag coefficient; (c) mean lift coefficient; and (d) r.m.s. lift coefficient.

that the Reynolds number and reduced velocity have some effects on the flow patterns. For Re ¼ 50 and 100, the wakes exhibit a 2S mode, while for Re¼150 and 200, a PþS mode appears at some lower Ur. Fig. 9 provides the instantaneous vorticity fields at selected Re and Ur. For Re¼ 50, vortices from the upstream cylinder cannot be released in the gap between two cylinders at Ur r10. As shown in Fig. 9(a) and (b), the flow separated from the upstream cylinder reattaches on the downstream one, generating a von Kármán vortex street of 2S mode in the wake; consequently, the two cylinders behave like a single bluff body, corresponding to the low-amplitude vibrations of the downstream cylinder. However, in general case, the vortices shedding from the upstream larger cylinder roll up in the gap region, then impinge on the downstream cylinder (Fig. 9(c)–(n)). Interestingly, as displayed in Fig. 9(g), (h), (k), and (l), the wakes behind the downstream cylinder develop into a PþS mode, which is identified by the observation that a vortex pair and a single vortex are shed at opposite sides of the downstream wake. The PþS mode was previously observed in the 2D simulations for the Re¼150 flow with free vibrations of two tandem equal-sized cylinders (Bao et al., 2012). In our work, since the validity of the 2D simulations may be questionable as the Re increases to a higher value especially for Re ¼200, more accurate 3D simulations remain to be done to check whether the P þS mode exists in reality. 4.4. Hydrodynamic forces Fig. 10 shows the drag and lift force statistics of the downstream cylinder versus Ur for different Re. As shown in Fig. 10(a), the negative mean drag coefficient (C D ) can be observed clearly at Ur r10 for Re¼50, due to the stagnant recirculation zone between two cylinders; with the advent of vortex shedding in the gap region, the C D increases sharply between Ur ¼10 and 11, then maintains a high level which is comparable to those for higher Re. The variations of C D show a similar trend for 100rRer200, where the flow dynamic mechanisms are completely dominated by the vortex–cylinder interaction. It is worth noting that all of the peak values of C D occur in the lock-in region, coinciding with the large-amplitude transverse vibration. As shown in Fig. 10(b), generally, higher Re yields larger r.m.s. values of the fluctuating drag coefficient C D' ; the same is true for C L' to be discussed later. Following a drop in the unlocked region, the C D' increases to reach a peak (or local peak) in the lock-in region. Fig. 10(c) shows that the values of the mean lift coefficient (C L ) are always near zero for Re¼50 and 100,

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Fig. 11. Frequency analyses of WIV of two unequal-sized cylinders for different Re (2-dof): (a) Re¼ 50; (b) Re¼ 100; (c) Re¼ 150; and (d) Re¼200.

since the time-averaged wake structures are symmetrical with regard to the wake centreline. In contrast, at some lower Ur for Re¼150 and 200, nonzero values of C L appear, corresponding to the occurrence of asymmetric PþS mode. For the lift fluctuation shown in Fig. 10(d), with the increasing of Ur, the C L' decreases to a minimum in the regime of lock-in, beyond which it recovers to a high level again. This strong lift fluctuation partially contributes to the amplified vibration of the downstream cylinder, even at a higher Ur.

5. Results of WIV with 2-dof 5.1. Frequency analysis and vibration responses For an elastically mounted cylinder immersed in a uniform flow, the dynamic interaction between the vortex shedding and the cylinder can also excite in-line vibration due to the drag force oscillating at a frequency which is double that of the vortex shedding. The frequency analysis for a 2-dof vibrating cylinder in the wake of a stationary larger cylinder is shown in Fig. 11, where the fox denotes the in-line vibration frequency. It is found that the vibration frequencies in the in-line direction are usually twice their counterparts in the transverse direction (fox ¼2foy). Nonetheless, at 7 rUr r9 for Re¼150 and 200, the vibration frequencies in both directions are found to be the same (fox ¼foy), resulting from the effect of the PþS vortex shedding mode to be presented later. When compared to the transversely-only case (see Fig. 4), the lock-in regions of the downstream cylinder with 2-dof are essentially consistent with the former situations for the same Re. Jauvtis and Williamson (2004) pointed out that if the structural mass ratio was less than 6.0, the in-line vibration had significant effects on the transverse response. Fig. 12 shows the maximum transverse and in-line amplitudes (Ymax/d, Xmax/d) of the downstream cylinder versus Ur for different Re. As observed in Fig. 12(a), the variation trends of Ymax/d are similar to those shown in Fig. 5(a). However, it is also noted that the in-line freedom induces an increase in the peak values of Ymax/d, which are approximately 10% larger than their counterparts for the transverse-only vibrations. Compared to Ymax/d, the Xmax/d is typically much smaller, as shown in Fig. 12(b). For Re¼ 50, the in-line vibrations with low amplitudes can be

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Fig. 12. Variations of the maximum vibration amplitudes versus Ur for different Re (2-dof): (a) transverse amplitude and (b) in-line amplitude.

Fig. 13. Variations of the mean in-line displacement of the downstream cylinder versus Ur for different Re (2-dof).

observed over all tested Ur, while for Re¼ 100 and higher, several remarkable vibrations appear, due to the in-line resonance where fox is near to fn. The variations of the mean in-line displacement (Xmean/d) versus Ur are shown in Fig. 13. The Xmean/d increases to larger values with the increase of Ur, except that when Ur r10 for Re¼50, it decreases to below zero. The appearance of negative Xmean/d can be ascribed to the negative drag force acting on the downstream cylinder. 5.2. Flow patterns Fig. 14 provides the flow pattern distributions behind the downstream cylinder subjected to 2-dof vibration. The dominant vortex shedding mode is still 2S for lower Re (Re ¼50 and 100). With the increase of Re up to 200, besides P þS, the C (chaotic) mode can also be observed within the lower Ur range. The fully developed vorticity fields at selected Re and Ur are shown in Fig. 15. It is noted that the vigorous in-line resonance leads to a dramatic change in the wake at (Re, Ur)¼ (200, 4), where a chaotic unstable vortex street forms behind the downstream cylinder. For each Re, although the rear cylinder moves further downstream with the increase of Ur up to 30.0, it still lies in the wake interference regime and experiences wake-induced vibration. However, due to the diminishing effect of the upstream wake, the transverse responses are slightly reduced compared to the transversely-only case at higher Ur for the same Re (see Fig. 5(a) and Fig. 12(a)). It is reasonable to infer that, at infinite Ur, the downstream cylinder would get rid of the wake interference, and behave as a single cylinder that vibrates with negligible amplitude. 5.3. Orbital trajectories The free vibrations of a single cylinder with 2-dof typically exhibit a figure-eight motion because the dominant frequency in the variation of drag is twice that of the lift (Prasanth and Mittal, 2009). Figs. 16–19 show the Lissajous figures of the downstream cylinder orbital trajectories in our WIV simulations. It can be observed that most of the cylinder orbits (including those for higher Ur which are not shown herein) appear to be a regular figure-eight trajectory. Besides, some

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Fig. 14. Flow patterns behind two unequal-sized cylinders for different Re and Ur (2-dof).

Fig. 15. Instantaneous vorticity fields for the fully developed flow past a stationary larger cylinder and an elastically mounted cylinder at selected Re and Ur (2-dof).

exceptions are also observed when the wakes behind the downstream cylinder present an asymmetric PþS mode. As can be seen from Figs. 18 and 19, at (Re, Ur)¼(150, 3), (150, 8), (200, 3), (200, 5) and (200, 8), the cylinder moves along distorted figure-eight trajectories; at (Re, Ur)¼(150, 7), (150, 9), (200, 7), and (200, 9), the vibrations present single-looped patterns,

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Fig. 16. Lissajous figures of the orbital trajectory of the downstream cylinder versus Ur for Re¼ 50.

Fig. 17. Lissajous figures of the orbital trajectory of the downstream cylinder versus Ur for Re¼ 100.

with the vibration frequencies in both transverse and in-line directions being the same. Another interesting observation is that the cylinder can move along figure-double eight trajectories in some cases, i.e., at (Re, Ur) ¼(50, 3), (100, 3), (100, 4), (150, 4) and (200, 4). This kind of orbital motion has not been reported yet in the existing literatures. In order to obtain a better understand, Fig. 20 provides the time histories and power spectra densities (PSD) of the cylinder displacements for two typical cases. At (Re, Ur)¼(150, 4) shown in Fig. 20(a), the cylinder experiences a steady periodic in-line vibration, while

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Fig. 18. Lissajous figures of the orbital trajectory of the downstream cylinder versus Ur for Re¼ 150.

Fig. 19. Lissajous figures of the orbital trajectory of the downstream cylinder versus Ur for Re¼ 200.

the transverse response becomes more complex since a third harmonic content can be clearly identified in the PSD (see the right of Fig. 20(a)). As shown in Fig. 20(b), beating is observed in the time histories of both transverse and in-line vibrations at (Re, Ur) ¼(200, 4), resulting in unclosed figure-double eight trajectories. This reflects in the PSD shown on the right of Fig. 20(b), which presents many spikes.

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Fig. 20. Time histories and power spectra densities of the displacements at (a) (Re, Ur) ¼(150, 4); and (b) (Re, Ur) ¼ (200, 4).

6. Conclusions The WIV responses of an elastically mounted cylinder with 1-dof and 2-dof behind a stationary larger cylinder have been numerically investigated at low Re. The two cylinders are in tandem arrangement with a center-to-center spacing of 5.5d. The Re changes from 50 to 200 and the Ur varies between 3.0 and 30.0 for each fixed Re. The four-step fractional finite element method, coupled with the ALE formulation is employed to solve the flow governing equations in a 2D moving mesh. According to the numerical results, major findings can be summarized as follows: For the 1-dof case, the transverse vibration frequency of the downstream cylinder is completely controlled by the vortex shedding of the upstream larger cylinder, and reduced to nearly half of the corresponding Strouhal number. The lock-in or synchronization for the downstream cylinder is observed. For Re ¼50, the lock-in covers a wide range of 11 rUr r17; as the Re varies from 100 to 200, it occurs in a relatively narrow band of reduced velocity around Ur ¼9. In most cases, the WIV dynamics are mainly dominated by the vortex–cylinder interaction mechanism. The transverse vibration amplitudes build up to a significantly higher level within and beyond the lock-in region when compared to the case of a single cylinder or two equal-sized cylinders. However, the transverse responses are suppressed at Ur r10 for Re¼50, which is ascribed to the shielding of the upstream larger cylinder. The Re and Ur have some effects on the flow patterns. For Re¼50 and 100, the dominant vortex shedding pattern takes on 2S mode, while for Re ¼150 and 200, the PþS mode appears within the lower Ur range. Since the validity of the 2D simulations may be questionable when the Re increases to a higher value (especially for Re¼200), more accurate 3D simulations remain to be done to check whether the PþS mode exists in reality. In the 2-dof situation, the downstream cylinder shares similar transverse response characteristics with the corresponding 1-dof case, but the peak amplitudes are approximately 10% larger than their counterparts for the latter. On the other hand, due to the diminishing effect of the upstream wake at higher Ur (when the rear cylinder moves further downstream), the transverse vibrations are slightly reduced. Generally, the in-line responses are of low amplitude; however, obvious vibrations can be observed at some Ur for ReZ100, which is related to the in-line resonance. The vigorous in-line resonant vibration is responsible for the formation of an unsteady C (chaotic) wake structure for Re¼200. Similar to the case of a 2-dof single

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cylinder, the motion trajectories of the downstream cylinder typically exhibit a regular figure-eight pattern. Additionally, because of the wake interference, some interesting orbital trajectories such as irregular figure-eight, figure-double eight and single-looped patterns also appear. Acknowledgment Financial support provided by the National Natural Science Foundation of China, China through Grant no. 51179101 is acknowledged. References Ahn, H.T., Kallinderis, Y., 2006. Strongly coupled flow/structure interactions with a geometrically conservative ALE scheme on general hybrid meshes. Journal of Computational physics 219, 671–696. Assi, G.R.S., Bearman, P.W., Meneghini, J.R., 2010. On the wake-induced vibration of tandem circular cylinders: the vortex interaction excitation mechanism. Journal of Fluid Mechanics 661, 365–401. Assi, G.R.S., Meneghini, J.R., Aranha, J.A.P., Bearman, P.W., Casaprima, E., 2006. 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