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2 Flow induced vibration of two rigidly coupled circular cylinders in tandem and side-by-side arrangements at a low Reynolds number of 5 Ming Zhao School of Computing, Engineering and Mathematics, University of Western Sydney, Locked Bag 797, Penrith, NSW 75, AUSTRALIA Tel: , m.zhao@uws.edu.au Abstract Flow induced vibration (FIV) of two rigidly coupled identical circular cylinders in tandem and sideby-side arrangements at a low Reynolds number of 5 is studied numerically. The two cylinders vibrate in the cross-flow direction and have the same displacement. The Navier-Stokes equations are solved by the finite element method and the equation of motion of the cylinders is solved by the fourth-order Runge-Kutta algorithm. Simulations are conducted for a constant mass ratio of and the gap ratios (defined as the ratio of the centre-to-centre distance between the two cylinders L to the cylider diameter D) of.5,, 4 and 6. The reduced velocities range from.5 to 5 with an increment of.5 for the tandem arrangement and.5 to 3 with an increment of.5 for the side-by-side arrangement. It is found that the gap between the two cylinders has significant effect on the response. For a tandem arrangement, the lock-in regime of the reduced velocity is narrower than that of a single cylinder for L/D=.5 and and wider than later for L/D=4 and 6. If the two cylinders are allowed to vibrate in the cross-flow direction, the vortex shedding from the upstream cylinder occurs at L/D as small as. The most interesting phenomenon found in the side-by-side arrangement is the combination of vortex-induced vibration (VIV) and galloping at L/D=.5 and. For L/D=.5 and, the response is dominated by VIV as V r <5 and by galloping as V r >5. At reduced velocities close to 5, the response is a combination of VIV and galloping.. Introduction Vortex-induced vibration (VIV) of a circular cylinder in fluid flow has been studied extensively because of its engineering importance. Most of the studies were focused on the fundamental case of

3 VIV of an elastically mounted rigid circular cylinder in steady flow. When an elastically mounted cylinder is placed in a fluid flow, large-amplitude oscillations occur when the vortex shedding frequency synchronizes with the oscillation frequency []. The synchronization between the vortex shedding and the vibration of the cylinder is also called the lock-in or lock-on in literature. Lockin occurs in a range of reduced velocity and the lock-in regime of the reduced velocity is dependent on the mass ratio and damping ratio of the system [, 3]. The mass ratio is defined as the ratio of the cylinder mass to the displaced fluid mass. The reduced velocity V r is defined as Vr U / f D, where n U is the free-stream velocity, f n is the structural natural frequency and D is the cylinder diameter. The natural frequency of free vibration of a cylinder in still water (defined as f nw in this study) is used in many laboratory studies of VIV to define the reduced velocity, i.e. V U f D [4, 5, 6]. In most of r / the numerical studies of VIV at low Reynolds numbers, the reduced velocity is defined using the structural natural frequency [7, 8]. At high mass ratios, the structural natural frequency and the natural frequency in fluid are essentially the same and the VIV frequency lock-in with the natural frequency [9, ]. When the mass ratio is in the order of O (), the response frequency can be significantly greater than the natural frequency measured in still water [6]. Williamson and Roshko [] found that the vortex-shedding pattern in the wake of a vibrating cylinder is related to the response mode of the cylinder. The number of the vortices in each cycle of vibration was used to define the vortex shedding mode. For example, the P mode stands for two pairs of vortices being generated in one cycle of vibration and S mode stands for two single vortices being generated in one cycle of vibration. For VIV of a circular cylinder in the cross-flow direction at low mass ratios, the vortex shedding is in the S mode in the initial branch and P mode in the upper and lower branch [6]. If a cylinder is allowed to vibrate both in the in-line and the cross-flow directions, the maximum response amplitude is increased to about.5 diameter of the cylinder and the velocity regime where the amplitude reaches its maximum is defined as super upper branch [5]. T mode nw

4 (two triplex vortices are shed from the cylinder in each vibration cycle) was observed when the amplitude is the maximum. Two circular cylinders close to each other in fluid flow have also attracted much interest. The interference between the two cylinders has significant effect on the wake vortex shedding flow [,, 3]. Williamson [4] found that for a certain range of gap between two cylinders in a side-byside arrangement, the wakes form the two cylinders were synchronized, either in phase or in antiphase with each other. When the distance between the centres of two side-by-side cylinder is less than.d, only one wake is formed [4-7]. Alam et al. [7] found that, as the spacing between two side-by-side circular cylinders is less than. diameters, the gap flow was biased toward one side, resulting in the formulation of a narrow wake behind one cylinder and a wide wake behind the other. For two cylinders in a tandem arrangement, the vortex shedding is found to occur only from the downstream cylinder if the distance between the two cylinders is less than a critical distance [8, 9, ]. This critical distance is about diameter depending on the Reynolds number. VIV of two circular cylinders is not studied as extensively as flow past two stationary cylinders or VIV of a single cylinder. Zdravkovich [] provided a review of the effects of the interference between two cylinders on the VIV. Fontaine et al. [] classified the response of two riser pipes in a tandem arrangement in two categories: Wake Induced Oscillation (WIO) and Vortex Induced Vibration (VIV). WIO, also referred to as galloping, a classical type of instability for a simplified description of the mechanism underlining the phenomenon and VIV is known to be a self-limited motion with a maximum order of diameter for a single riser []. The galloping in the case of two cylinders in tandem is also referred to be interference galloping [3] or wake galloping [4]. Bokaian and Geoola [5] studied vibration of an elastically mounted circular cylinder in the wake of a fixed upstream cylinder and found that the response of the cylinder can be a VIV, a galloping or a combination of VIV and galloping, depending on the gap between the two cylinders. Numerical studies on VIV of two cylinders in tandem were mainly conducted at relatively low Reynolds 3

5 numbers and gaps between the two cylinders of 5 or 5.5 diameters [6, 7, 8]. Zang et al. [9] and Zhao and Yan [3] investigated VIV of two cylinders of different diameter close to each other and found the interference between the two cylinders has significant effect on the VIV. The studies of VIV of two side-by-side cylinders in fluid flow are fewer than those of two tandem cylinders. Huera-Huarte and Gharib [3] investigated vortex-induced vibration of two flexible cylinders in side-by-side and tandem arrangements. It was found that the interference between the two cylinders was very weak if the centre-to-centre gap exceeded 3.5 times the cylinder diameter in the side-by-side configuration. Wang et al. [3] focused on their study on the effects of turbulence on the response of two side-by-side cylinders in flow. It was found that the enhancement of the response by the turbulence is significant when the two cylinders are very close to each other. (a) y L (b) y D Cylinder Cylinder Cylinder Flow D O D x Flow O L x D Cylinder Fig. Sketch of flow past two cylinders in (a) tandem and (b) side-by-side arrangements In this study, VIV of two rigidly coupled cylinders in tandem and side-by-side arrangements is studied by numerical method. Since the two cylinders are rigidly coupled, they vibrate as a single body. In offshore oil and gas engineering, two pipelines or riser pipes are sometimes strapped together at a certain intervals, forming a rigidly coupled two-pipe system. In most engineering applications of cylindrical structures in fluid flow, the Reynolds numbers are in the threedimensional turbulent regime. The two-dimensional model at a low Reynolds number is used in this study based on the following considerations. Firstly, two-dimensional simulations allow extensive 4

6 numerical simulations under various flow conditions to be conducted at affordable computational time. Secondly, two-dimensional simulations at laminar flow regime can identify the effects of the gap between the two cylinders and the flow directions on the response without any influence from the three-dimensionality. The outcomes found in this study can be guidance for the future threedimensional simulations. Two-dimensional (D) Navier-Stokes equations are solved to simulate the flow at a Reynolds number of 5. The D model is appropriate for the Reynolds number of 5 because the wake flow at this Reynolds number of 5 is two-dimensional and laminar [33, 34].. Governing equations and numerical method Fig. shows the sketch of two rigidly coupled cylinders in tandem and side-by-side arrangements. The two cylinder system is elastically mounted with zero structural damping. The cylinder system is allowed to vibrate in the cross-flow direction only. The structural damping is assigned to be zero to obtain the most intense vibrations [35]. The two cylinders vibrate as a whole and have same displacement and frequency. The two-dimensional incompressible Navier-Stokes (NS) equations are the governing equations for simulating the fluid flow. The Arbitrary Lagrangian Eulerian (ALE) scheme is applied to deal with the moving boundaries of the cylinder surface. The non-dimensionalization method of the NS equations is the same as that used by Zhao et al. [8]. The velocity (u, v), the time t, the length (x, y) and the pressure p are nondimensionalized as ( u, v) ( u ~, v ~ ) /( fnd), t ~ t fn, ( x, y) ( ~ x, ~ y ) / D, p ~ p /( f D n ), respectively, where the tilde denotes the dimensional parameters, ρ is the fluid density and f n is the structural natural frequency of the two cylinder system. The non-dimensional ALE formulation of the NS equations for incompressible flows are expressed as ui t u j u j,mesh u x j j p x i Vr ui, () Re x x j j 5

7 ui x i, () where x =x and x =y are the Cartesian coordinates in the in-line and the transverse directions of the flow, respectively; u i is the fluid velocity component in the x i -direction; t is the time and u i, mesh is the velocity of the movement of the mesh nodes. The nondimensionalization method used in this study leads to a nondimensional free-stream velocity that is equal to the reduced velocity and a nondimensional vibration frequency that is equal to the ratio of the vibration frequency to the structural natural frequency. The nondimensional equation of the motion of the cylinder is Y t Y 4 t 4 Y V C m r L * (3) where Y is the displacement of the cylinder system in the cross-flow direction, m * m / m is the d mass ratio with m being the mass of the cylinder and m d being the displaced mass of the fluid; c / km is the damping ratio with c and k being the damping constant and spring constant of the system, respectively; C L is the lift coefficient of the cylinder system defined by C L FL /( DU / ) with F L being the sum of the lift forces on the two cylinders. The equations of motion Eq. (3) is solved using the fourth-order Runge-Kutta algorithm. The two-dimensional Navier-Stokes equations are solved by the Petrov-Galerkin Finite Element Method (PG-FEM) [36]. After each computational time step in the numerical simulation, the positions of the two cylinders change and each finite element node needs to be moved accordingly. The governing equation for calculating the displacements of the nodes of the FEM mesh is S, (4) y where, S y represents the displacement of the nodal points in the y-direction; γ is a parameter that controls the mesh deformation. In order to avoid excessive deformation of the near-wall elements, the parameter γ in an finite element is set to be / A, with A being the area of the element [36, 37]. By specifying the displacements at all boundaries, Eq. (4) is solved by a Galerkin FEM. The 6

8 displacement S y is the same as the displacement of the cylinders on the cylinder surfaces and zero on the rest of the boundaries. Fig. Computational mesh near the two cylinders A nondimensional rectangular computational domain of a length of 5 (in the flow direction) and a width of 4 (in the cross-flow direction) is discretized into quadrilateral four-node linear finite elements. Fig. shows examples of finite elements around the cylinders. Zhao et al. [38] found that if the width of the computational domain (W) to the cylinder diameter ratio (W/D) is greater than diameter, the effects of the side boundaries on the results has been negligibly weak. In this study, W/D=8 for a side-by-side arrangement and 4 for a tandem arrangement. The centre of the two cylinder system is located from the inlet boundary and equal distance to the two lateral boundaries of the computational domain. Initially the velocity and the pressure in the whole computational domain are zero. The velocity at the inlet boundary is V r and the velocity at the cylinder surfaces equals to the vibration speed of the cylinder. At the outflow boundary the pressure is zero and the gradient of the velocity in the flow direction is zero. At the two lateral boundaries, the gradient of the pressure and the velocity in the cross-flow direction are zero. 3. Validation of the numerical model 7

9 The numerical model has been validated by Zhao et al. [8] and used to simulate VIV of a square cylinder in steady flow at a low Reynolds number of. In this study, further validation is conducted for the cases of flow past two stationary cylinders at Re=, VIV of a circular cylinder at Re=5 and VIV of a circular cylinder in the wake of a stationary circular cylinder at Re=5. Flows past two stationary cylinders in a tandem arrangement at gap ratios of L/D=.5, and 4 and a Reynolds number of are simulated and the results are compared with the published data in Table. Meneghini et al. [8] simulated flow past two cylinders in a tandem arrangement at Re= and L/D=.5 to 4. Table shows the comparison of the mean drag coefficient C D, the amplitude of the lift coefficient A CL and the Strouhal number S t. The subscripts and stands for the upstream and the downstream cylinders, respectively throughout the paper. Only the results for L/D=.5 and 4 are available in Borazjani and Sotiropoulos [35]. The A CL and A CL are defined as the amplitudes of the lift coefficient of the upstream and the downstream cylinders, respectively. Because A CL and A CL by Meneghini et al. [8] are measured from the time histories of the force coefficient in their paper, there is certain uncertainty in them. The results from the three models in Table for L/D=.5 are very close to each other. Flow past a stationary single cylinder at Re= is simulated and the mean drag coefficient and the Strouhal number are C D =.336 and S t =.94, respectively. The corresponding values calculated by Meneghini et al. [8] are C D =.3 and S t =.96, respectively. The drag coefficient of the upstream cylinder is positive and that of the downstream cylinder is negative when L/D is small. When L/D is small, the flow pattern is characterized by two quasi-steady vortices in the gap region and periodic vortex shedding from the downstream cylinder. The drag coefficient and the Strouhal number at small L/D are smaller than their corresponding values for a single cylinder. The oscillation of the lift coefficient of both cylinders increases significantly as L/D is increased from to 4. The lift coefficients are increased because the vortex shedding from the upstream cylinder occurs at large L/D. Obvious difference occurs among the three sets of results as L/D=4. The present results agree will those by Meneghini et al. [8] but have big difference from 8

10 those in Ref. [35]. Borazjani and Sotiropoulos [35] stated that the difference occurs because the spacing ratio L/D=4 is close to the critical spacing where the flow patterns undergo an abrupt change. Table Comparison of the numerical results of flow past two stationary cylinders in tandem Reference L/D C D A CL S t C D A CL S t Borazjani and Sotiropoulos [35] Meneghini et al. [8] Present Meneghini et al. [8] Present Borazjani and Sotiropoulos [35] Meneghini et al. [8] Present The second validation case is the VIV of a circular cylinder at low Reynolds number of Re=5, low reduced mass of M red = and zero damping ratio. It should be noted that the reduced mass, which is defined as M red m / D, is different from the mass ratio used in this study. In this example M red is used in order to compare the numerical results with other published data under the same calculation parameters. Fig. 3 (a) shows the comparison between the calculated response amplitudes with the numerical results in three published papers. The amplitude is defined as ( Y max Ymin ) /. The variations of the response amplitude with the reduced velocity from different models are similar to each other. The present results are closer to the results by Bao et al. [39]. Fig. 3 (b) shows the comparison of the calculated mean drag coefficient, root mean square (RMS) drag and lift coefficients with their counterparts calculated by Bao et al. [39]. The agreement between the two sets of results are very good except the mean drag coefficient at V r =4, where the vibration amplitude is the maximum. In the present results, all the mean drag, RMS drag and RMS lift coefficients, and the response amplitude reach their maximum values at V r =4. The numerical study by Singh and Mittal [7] of DOF VIV and the experimental study by Khalak and Williamson [4] of DOF VIV in the cross-flow direction also showed that the mean drag coefficient, the RMS drag and lift coefficients A y 9

11 reached their maximum values at a same reduced velocity. In the results by Bao et al. [39], the response amplitude and the lift coefficient at V r =4 are the maximums, while the mean drag coefficient and the RMS drag coefficient at V r =4 are not. The validation of the numerical model on the interference between two cylinders on VIV is testified by comparing the numerical results with those by Bao et al. [39], who studied flow interference between a stationary cylinder and an elastically mounted cylinder arranged in a tandem arrangement. Vibration of an elastically mounted circular in the wake of an identical stationary cylinder at Re=5 is simulated. The gap ratio L/D, the reduced mass and the damping ratio of the downstream cylinder is the same as those used by Bao et al. [39], i.e. L/D=4, M red = and ζ=. Fig. 4 shows the variations of the response amplitude and the root mean square lift coefficient with the reduced velocity. The experimental data by Assi et al. [4] are also plotted in Fig. 4 (a) for comparison. In the experiments, the mass ratio was (equivalent to M red =.57), which is slightly lower than that used in the numerical simulation and the Reynolds numbers are in the turbulent flow regime in range of 3 3. The present results of both response amplitude and the lift coefficient agree exceptionally well with their counterparts calculated in Ref. [39]. The lock-in regime of the response for the two cylinder system in Fig. 4 is much wider than that of a single cylinder. Both numerical results agree well with the experimental data before the response amplitude reaches its maximum (V r <6). As V r >6, the response amplitude in the experiments continues increasing with the increasing V r while that from the numerical simulation decreases. The discrepancy between the experimental data and the numerical results is attributed to the difference in the Reynolds number.

12 .8 (a) Responseamplitude Ahn and Kallinderis [4] Borazjani and Sotiropoulos [35] Bao et al. [39] Present.6 A y /D.4. Force coefficient (b) Force coefficient Mean CD, Bao et al. [39] Mean CD, Present RMS CL, Bao et al. [39] RMS CL, Present RMS CD, Bao et al. [39] RMS CD, Present Fig. 3 Variations of the response amplitude and the force coefficients with the reduced velocity for a single cylinder at Re=5, Mred= and ζ=. Ay/D (a) Response amplitude of the downstream cylinder Bao et al. [39] Present Assi et al. [4] m *, m * ζ 3, Re= V r Fig. 4 The response amplitude and the root mean square lift coefficient of an elastically mounted cylinder in the wake of stationary cylinder with L/D=4, Mred= and ζ= V r C L,rms.5.5 (b) Root mean square of the lift coefficient of the downstream cylinder V r Bao et al. [39] Present

13 4. VIV of two cylinders in the tandem arrangement VIV of two rigidly coupled circular cylinders in the tandem arrangement at gap ratios of L/D=.5,, 4 and 6 and reduced velocities in the range of.5 to 5 with an increment of.5 are simulated. Flow past two stationary cylinders in the tandem arrangement at Re=5 is firstly simulated and the variations of the mean drag ( C ) and the root mean square (RMS) lift coefficients D of both cylinders with the gap ratio L/D are shown in Fig. 5. The variation of mean drag coefficient C D of the downstream cylinder with L/D by Zdravkovich and Pridden [4] is plotted in Fig. 5 (a) for comparison. The mean drag coefficient on the downstream cylinder is negative at L/D=.5 and, because vortex shedding from the upstream cylinder does not occur in such small gaps. At small gaps, the shear layers that are separated from the two sides of the upstream cylinder reattach to the downstream cylinder, instead of forming vortex shedding flow in the gap between the two cylinders. Menghini et al. [8] reported that the force could be negative at small gap ratios because the downstream cylinder was inside the wake formed by the upstream cylinder where the pressure is low. Zdravkovich and Pridden [4] measured the pressure for flow past two cylinders in tandem and confirmed that the pressure in front of the downstream cylinder was lower than that downstream at low gap ratios, leading to a negative drag force. As the gap ratio is increased to L/D=4, the vortex shedding from the upstream cylinder occurs. The difference between the mean drag coefficient of the upstream cylinder and that of a single cylinder is small. The RMS lift coefficient on both cylinders are extremely weak at L/D=.5 and because of two reasons. Firstly the vortex shedding does not occur in the gap between the two cylinders at small gap ratios. Secondly, the location, where interaction between the free shear layers from the two sides of the downstream cylinder occurs, is far downstream of the cylinders, making very small influence on the lift forces on the two cylinders. Once L/D is 4, the RMS lift coefficient on the downstream cylinder is significantly greater than that of a single cylinder, while the RMS lift coefficient of the upstream cylinder is almost the same as

14 that of a single cylinder. The RMS lift coefficient is greater than that of a single cylinder at large gap ratios because of the influence of the unsteady vortex shedding flow from the upstream cylinder. Zdravkovich and Pridden [4] found that the flow changes modes at about L/D=3.5. When L/D<3.5, there is almost no gap flow and the drag on the downstream cylinder is negative. As L/D>3.5, the vortex shedding from the upstream cylinder is almost fully developed and impinges on the downstream cylinder. Meneghini et al. [8] found that the critical L/D between the two flow patterns is L/D=3 for a low Reynolds number of Re=. 3.5 (a) Mean Drag coefficient Upstream cylinder Downstream cylinder Single cylinder Exp., downstream cylinder, Re=3 [4].5 (b) RMS lift coefficient Upstream cylinder Downstream cylinder Single cylinder CD.5 C L,rms L/D L/D Fig. 5 Variations of the mean drag and root mean square lift coefficients with L/D for two stationary cylinders in the tandem arrangement Fig. 6 shows the time histories of the vibration displacement of the two cylinders in the tandem arrangement. At L/D=.5, the vibrations of all the reduced velocities are perfectly periodic. The amplitude reaches its maximum at V r =4.5. Beating is the typical phenomenon occurring at V r =4.5 to 6 for L/D=. The beating phenomenon was also found in the numerical simulation of VIV of a single cylinder at reduced velocities where the response amplitude was reaching its maximum [7, 43]. The beating period increases with an increase in the reduced velocity. In each beating period, the vibration amplitude increases slowly with time and then quickly reduces to very small amplitude followed by another beating period. The vibration histories for L/D=4 are regular except with slightly variation in the amplitude as shown in Fig. 6 (h). For L/D=4, the vibration at V r =6 shown in Fig. 6 3

15 (g) has the largest amplitude. For L/D=6, each time histories between 9 V 3. r 5 comprises two frequency components as shows Fig. 6 (k) and (l), with one frequency being about twice another..4. (a) L/D=.5,Vr= (b) L/D=.5,Vr= (c) L/D=,Vr= (d) L/D=,Vr= (e) L/D=,Vr= (f) L/D=,Vr= (g) L/D=4,Vr= (h) L/D=4,Vr= (i) L/D=6,Vr= (j) L/D=6,Vr= (k) L/D=6,Vr= (l) L/D=6,Vr= tfn tfn Fig. 6 Time histories of the vibration displacement for the tandem arrangement Fig. 7 shows the amplitude spectra of the vibration for some typical cases, which are obtained based on the Fast Fourier Transform (FFT). The spectra of the vibration displacement and the lift coefficient are plotted in the same figure in order to see the link between the vibration and the lift force. The spectra of the lift coefficient are based on the total lift force, i.e. the sum of the lift forces on the upstream and downstream cylinders. Fig. 7 (a) and (b) are typical regular periodic vibrations with single peak frequencies. Fig. 7 (c) and (d) are the spectra corresponding to the beating shown in Fig. 6 (d) and (e), respectively. In Fig. 7 (c) the spectrum comprises a number of peaks and the secondary peak frequency is very close to the primary peak frequency (the peak frequency with the 4

16 highest amplitude). In Fig. 7 (d), the peak frequencies of the displacement and the lift coefficient are not as distinct as those in Fig. 7 (c). The energy at the frequencies close to the primary peak frequency leads to the beating. Each spectrum in Fig. 7 (e) and (f) has two peaks and the amplitudes corresponding to these two peaks are comparable with each other. The lift coefficient corresponding to the lower peak frequency is smaller than that corresponding to the higher peak frequency. However, the lower peak frequency has higher amplitudes because it is closer to the natural frequency. All the spectra shown in Fig. 7 are in the lock-in regime, where the vortex shedding frequency locks onto the vibration frequency. The frequency of the highest peak in the displacement spectrum is the same as that in the lift coefficient spectrum except in Fig. 7 (e) and (f).. (a) L/D=.5, Vr= (b) L/D=.5, V r =5.8.4 (c) L/D=, V r =5.5 Ay/D. Y CL.6.4. A CL A y /D. Y CL.6.4. A CL Ay/D.3.. Y CL.5 A CL 3 f/f n. (d) L/D=, V r =6.3 3 f/fn. (e) L/D=6, V r =9.4 3 f/f n. (f) L/D=6, V r =.4 Ay/D. Y CL.. A CL A y /D. Y CL.3.. A CL Ay/D. Y CL.3.. A CL f/f n f/f n f/f n Fig. 7 Amplitude spectra of the vibration based on the FFT analysis Fig. 8 shows the variation of the response amplitude with the reduced velocity. The results for the four calculated gap ratios are plotted separately in Fig. 8 (a) and (b) in order to avoid the unclearness due to too many curves and symbols in one figure. The results for a single cylinder are also plotted in the figure for comparison. Fig. 9 shows the variation of the response frequency with the reduced velocity. If the vortex shedding frequency and the vibration frequency follows the Strouhal law, vibration frequency should be a linear function of the reduced velocity. For a single cylinder, the vortex shedding frequency locks onto the vibration frequency of the cylinder instead of following the Strouhal law when synchronization occurs [9]. At high mass ratios in the order of O(), the vibration frequency is the same as the natural frequency of the cylinder [6, 44]. 5

17 However, at low mass ratios, the vibration frequency at lock-in was found to be slightly greater than the natural frequency of the cylinder [4, 6, 45]. The response amplitude in the lock-in regime is significantly higher than those outside the lock-in regime. It can be seen in Fig. 8 that the response amplitude for a single cylinder is increased suddenly as the reduced velocity is increased from 3 to 3.5. The vibration frequency as V 3 exactly follows a linear function of the reduced velocity. As r V r 4, the response frequency for a single cylinder increases slowly with the increasing V r until V r =6 and keeps almost constant in the regime of 6 V r 8. The amplitudes in the regime of 3.5 V r 8 are significant greater than those in other reduced velocities. The lock-in regime for a single cylinder is believed to be 4 V r 8..8 (a) Amplitude for L/D=.5 and Tandem, L/D=.5 Tandem, L/D=.6 Single cylinder Ay/D (b) Amplitude for L/D=4 and 6 Tandem, L/D=4 Tandem, L/D=6.6 Single cylinder Ay/D V r Fig. 8 Variation of the response amplitude with the reduced velocity for the tandem arrangement 6

18 3.5 (a) Frequency for L/D=.5 and Tandem, L/D=.5 Tandem, L/D= Single cylinder f/fn.5 f/f n =.5 f/fn (b) Frequency for L/D=4 and 6 Tandem, L/D=4 Tandem, L/D=6 Single cylinder f/f n = Fig. 9 Variation of the response frequency with the reduced velocity for the tandem arrangement V r It can be seen in Fig. 8 and Fig. 9 that the lock-in regime for two cylinders in tandem is significantly different from that of a single cylinder and is dependent on the gap ratio L/D. The lockin regime is the narrowest at L/D=.5. The variation of the response frequency f/f n with the reduced velocity for L/D=.5 is almost a linear function except at V r =4.5, 5 and 5.5, where f/f n is slightly deviated from the linear function. The response amplitude for L/D=.5 reaches its maximum at V r =4.5. As V r 6, the response amplitude for L/D=.5 has been very small and changes little with the increasing V r. Based on Fig. 8 (a) and Fig. 9 (a), the lock-in regime for L/D= is 3 V r 6, which is also narrower than that of a single cylinder. The response frequency for L/D= in the lock- 7

19 in regime is smaller than the structural natural frequency, i.e. f/f n <. The maximum response amplitude in the lock-in regime for L/D= is much higher than that for L/D=.5. At L/D=4, the clear cut lock-in regime is hard to be identified in Fig. 9 (b) because the vibration frequency increases smoothly with the increasing V r. There is not any disconnection in the variation of the response amplitude with the reduced velocity either. Taking a close look at the frequency v.s. reduced velocity curve in Fig. 9 (b), it can be seen that the response frequency is a linear functions of the reduced velocity in the reduced velocity regimes of.5 V r 3. 5 and 8.5 V r 5. In the regime 8 of 4 V r 8, the increase rate of the response frequency for L/D=4 with the increasing V r is much smaller than those in other ranges of V r. It can be seen in Fig. 8 (b) that the response amplitudes in the range of 4 V r 8 are high. The regime of 4 V r 8 is treated to be lock-in regime for L/D=4 in this study. The vibration amplitude as V r >8 decreases gradually with the increasing V r and is still above.35 at V r =5. In Fig. 8 (b) there are two maximum values in the response amplitude curve for L/D=6. One is at V r =4 and another one is at V r =7. The response frequency is a linear function of the reduced velocity as V r 3 and it increases suddenly as V r is increased from 3 to 3.5. A sudden decrease in the response frequency occurs as the reduced velocity is increased from 4.5 to 5. The nondimensional response frequency f/f n increases from a value slightly less than to a value slightly greater than as the reduced velocity is increased from 5 to 8.5. The response amplitude for L/D=6 increases with the increasing V r until V r =7 after which the response amplitude almost keeps constant until V r =8.5. The response frequency decreases suddenly to a value slightly less than as the reduced velocity increases from 8.5 to 9 and then increases slowly with the increasing reduced velocity until V r =3.5, where f/f n is about. Although the response frequency f/f n in the reduced velocity regime of 9 V 3.5 r is close to, the response amplitude is less than. in this region as shown in Fig. 8 (b). The reason of the low response amplitude at frequencies close to the natural frequency can be explained by examine the spectra as shown in Fig. 7 (e) and (f), where the total lift coefficient of the

20 cylinder system has two predominant frequencies with one being close to the natural frequency and another one being not. The frequency component of the lift coefficient that is close to the natural frequency is secondary compared with the one with the higher amplitude. The secondary component of the force generates higher amplitude than the primary component because it synchronizes with the natural frequency. However, the amplitude A y /D is still less than. because force corresponding to the vibration frequency is the secondary component. The regime of 9 V 3. r 5 is not included in the lock-in regime because the vortex shedding frequency which is the same as the leading frequency of the lift coefficient is not the same as the vibration frequency of the cylinder system and the response amplitude in this range of reduced velocity is very small. So, the whole lock-in regime for L/D=6 is 3.5 V r Fig. shows the variation of the RMS lift coefficient of each individual cylinder and the RMS lift coefficient based on the total force (sum of the lift forces on both cylinders) with the reduced velocity. The lift coefficient based on the total force is defined as the total lift coefficient. The RMS lift coefficient on the upstream cylinder is generally smaller than that on the downstream cylinder because the downstream cylinder is influenced by the vortex shedding flow in the gap. It can be seen by comparing Fig. 8 (a) with Fig. (c) that in the initial branch of the response (where the response amplitude increases with the increasing V r ), the RMS lift coefficient of the total force also increases with the increasing V r as the gap ratio L/D=.5 and. This is similar to that of a single cylinder case reported in Ref. [4]. For L/D=.5 and, the RMS lift coefficient peaks at the reduced velocity where the response amplitude peaks. Different from those as L/D=.5 and, the RMS total lift coefficient for L/D=4 and 6 reaches its minimum value when the response amplitude peaks. The variation of RMS total lift coefficient with the reduced velocity as L/D=4 and 6 is similar to the case where the upstream cylinder is fixed and the downstream cylinder is free to vibrate (as shown in Fig. 4), because in these two gaps, the vortex shedding from the gap has significant effects on VIV. The total lift coefficient at the highest calculated reduced velocity for L/D=3 is significantly greater than those 9

21 at other gaps. This leads to big response amplitudes even if the lock-in does not occur as shown in Fig. 8 (b). (a) C L,rms of the upstream cylinder CL,rms.5 Tandem, L/D=.5 Tandem, L/D= Tandem, L/D=4 Tandem, L/D=6.5 C L,rms (b) C L,rms of the downstream cylinder Tandem, L/D=.5 Tandem, L/D= Tandem, L/D=4 Tandem, L/D= (c) C L,rms of the total force.5 Tandem, L/D=.5 Tandem, L/D= Tandem, L/D=4 Tandem, L/D=6 CL,rms V r Fig. Variation of the RMS lift coefficient with the reduced velocity for two cylinders in the tandem arrangement

22 5 (c) Phase between Y and C L for L/D=.5 and 8 35 Tandem, L/D=.5 ψ (degree) 9 45 Tandem, L/D= (c) Phase between Y and C L for L/D=4 and 6 ψ (degree) 35 9 Tandem, L/D=4 Tandem, L/D= V r Fig. Variation of the phase between the response displacement and the total lift force with the reduced velocity for tandem arrangement Fig. shows the variation of the phase between the response displacement and the total lift force (ψ) with the reduced velocity for the tandem arrangement. It was found that, for a single cylinder, the phase between the response displacement and the lift force jump from to 8 as the response transits from upper branch to lower branch [4]. Similar to that of the single cylinder case, the phase ψ jumps from to 8 at V r =6, 6.5 and 6.5 for L/D=.5, and 4, respectively. It can be seen by comparing Fig. 8 and Fig. that the jump of the phase occurs after the response amplitude reaches its maximum value. It is interesting to see that the phase ψ for L/D=6 jumps to 8 at V r =7.5 and jumps back to at V r =9. It jumps from to 8 again at V r =.5. The phase ψ becomes again in the range of 9 V r, where the response frequency approaching natural frequency as seen in Fig. 9 (b). In the reduced velocity range of 9 V r, both the response

23 displacement and the lift force have double frequencies as shown in Fig. 7 (e) and (f), with the lower frequency being closing to the natural frequency. It is found that phase ψ between the response displacement and the lift force corresponding to the lower frequency is about and that corresponding to the higher frequency is about 8. Fig. (b) only shows the phase corresponding to the lower frequency for L/D=6 and 9 V r since the response amplitude at this frequency is higher than another. Because the lift coefficient is related to the vortex shedding, the phase between the response displacement and the lift coefficient reflects the relation between the response and the vortex shedding flow. For example, it is found that if the response displacement is in phase with the lift coefficient, whenever the cylinders reach their lowest position, the vortices below the cylinders are stronger than those above the cylinders, leading to negative lift coefficient. CL,rms CL CL CL+CL (a) (b) (c) (d) C L tf n Fig. Vorticity contours around two cylinders in a tandem arrangement at L/D=.5 and Vr=4.5 The vortex shedding flow pattern is examined by observing the contours of the vorticity around the cylinders. The vorticity is defined as v / x u / y. Fig. shows the vorticity contours around two cylinders in the tandem arrangement at L/D=.5 and V r =4.5, where the amplitude is the maximum. The four instants for the vortex shedding flow structures are marked in the time histories

24 of the cylinder displacement and the lift coefficients, which are shown at the bottom of Fig.. Periodic vortex shedding is observed and two vortices are found to be shed in one cycle of vibration. The vortex shedding pattern in Fig. is very similar to that of two stationary cylinders in tandem at the same gap ratio. The vortices are only shed from the downstream cylinder, while the fluid velocity in the gap between the two cylinders is very weak. The two cylinders behave as a single body which is more streamlined. The hydrodynamic lift coefficient on the upstream cylinder is very small since there is not vortex shedding behind it. The total lift coefficient CL CL is in-phase with the vibration displacement, resulting large response amplitude. C L,rms CL CL CL+CL 4 (a) (b) (c) (d) (e) (f) tf n Fig. 3 Vorticity contours for two cylinders in a tandem arrangement at L/D= and Vr=5 CL 3

25 Fig. 3 shows the vorticity contours around two cylinders in the tandem arrangement at L/D= and V r =5. For two stationary cylinders in tandem, it has been found that the vortex shedding from the upstream cylinders does not occurs if the gap ratio L/D is less than about 3.5 or 4 [4, 8]. The critical gap for the onset of the vortex shedding from the upstream cylinder at low Reynolds numbers is slightly greater than that at high Reynolds numbers. At L/D=, the vortex shedding does not occur behind the upstream cylinder if the two cylinders are stationary. This can be evidenced by the extremely small RMS lift coefficients on both cylinders as shown in Fig. 5 (b). However, it can be been in Fig. 3 that the vortex shedding behind the upstream cylinders occurs at a gap ratio as small as L/D=. In Fig. 3 (b), where the cylinders are moving downwards, the negative vortex from the top side of the upstream cylinder separates from the cylinder surface. This vortex is shed from the cylinder because the relative direction of the fluid flow to the cylinders is biased upwards due to the downward movement of the cylinders. This allows the vortex from the upstream cylinder being shed from the top-right direction. The negative vortex that is shed in Fig. 3 (b) scratches the top side of the downstream cylinder as seen in Fig. 3 (c). The lift force on the downstream cylinder is very small because the negative vortex coming from the top side of the upstream cylinder cancels the downward forces by the positive vortex from the bottom of the downstream cylinder. It appears that the hydrodynamic force on the upstream cylinder dominate the vibration. For L/D=, the vortex shedding from the upstream cylinder is only observed in the lock-in regime, where the response amplitude is large. Outside the lock-in regime, the vortex shedding only occurs in the wake of the downstream cylinder and the fluid velocity in the gap is very small, which is similar to that in Fig.. The beating, which is a typical phenomenon for L/D= and V r =5 as shown in Fig. 6 (d), is found to be related to the change in the vortex shedding. It is found that the vortex shedding from the upstream cylinder disappear and reappear periodically. Whenever the vortex shedding from the upstream cylinder is suppressed, the response amplitude is very small as shown in Fig. 3 (e) and (f). 4

26 The period of beating in the time history shown in Fig. 6 (d) is the same as the period of the disappearance and reappearance of the vortex shedding from the upstream cylinder. Fig. 4 shows Vorticity contours for two cylinders in a tandem arrangement at L/D=4 and V r =6, where the vortex shedding from the upstream cylinder is fully developed. After each vortex is shed from the upstream cylinder, it impinges on the downstream cylinder and affects the vortex shedding from the downstream cylinder significantly. Fig. 4 (a) and (c) shows that each vortex from the upstream cylinder is split into two parts by the downstream cylinder. The lift coefficients on the two cylinders are almost in anti-phase with each other. In Fig. 4 (b) the lift coefficient on the upstream cylinder is positive because the stagnation point of the flow is located on the bottom-left side of the upstream cylinder and the flow relative to the cylinder is in the top-right direction. However, the lift coefficient on the downstream cylinder in Fig. 4 (b) is negative because of attraction of the strong positive vortex attaching to the bottom surface of the cylinder. After the positive vortex that is shed from the upstream cylinder in Fig. 4 (a) attacks the downstream cylinder, most of this vortex is covected to the bottom side of the downstream cylinder and resulting strong positive vortex from at the bottom surface of the downstream cylinder in Fig. 4 (b). 5

27 CL,rms CL CL CL+CL (a) (b) (c) (d) - C L tf n Fig. 4 Vorticity contours for two cylinders in a tandem arrangement at L/D=4 and Vr=6 The vortex shedding pattern is found affected not only by the gap ratio L/D but also the reduced velocity. Fig. 5 shows vorticity contours for two cylinders in a tandem arrangement when the cylinders are at the lowest location for L/D=4 at four reduced velocities. At V r =4, each negative vortex that has been totally separated from the upstream cylinder will be convected by the flow for half period of time before it reaches the downstream cylinder. It passes through the top side of the downstream cylinder. With the increase in the reduced velocity, the shear layers from the upstream cylinder extend further downstream and the vortices attack the downstream cylinder right after they are shed from the upstream cylinder. The vortex shedding from the upstream cylinder is always in the S mode, i.e. two single vortices are shed from the upstream cylinder in each vibration period. Singh and Mittal [7] also found that the S vortex shedding mode in the wake of an elastically mounted cylinder at low Reynolds numbers less than 3. 6

28 Fig. 5 Vorticity contours for two cylinders in a tandem arrangement when the cylinders are at the lowest location for L/D=4 and Vr=4, 4.5, 5 and 7 Fig. 6 shows the vorticity contours for two cylinders in a tandem arrangement at L/D=6 and V r =4. The influence of the vortex shedding from the upstream cylinder on that of the downstream cylinder is very similar to that in Fig. 5 (a). Each vortex that is shed from the upstream cylinder reaches the downstream cylinder after about one vortex shedding period. Instead of impinging on the front surface of the downstream cylinder, the vortices from the upstream cylinder scratch the top and the bottom surfaces of the downstream cylinder. The two rows of vortices from the upstream cylinder attract the vortices that are shed from the downstream cylinder upwards and downwards, respectively. The vortex street downstream the downstream cylinder is the combination of the vortex shedding from both cylinders. Fig. 7 shows Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and V r =8. The vortex shedding in Fig. 7 is similar to that in Fig. 5 (c) and (d). It can be seen that the vortex shedding in Fig. 7 (a) is not exactly the same as that in Fig. 7 (c). The positive vortex that has been shed from the upstream cylinder in Fig. 7 (a), where the cylinders are in their highest position, attacks the front surface of the downstream cylinder. However, in Fig. 7 (c), where the two cylinders are in their lowest position, the negative vortex in the gap has not reached the downstream cylinder. This negative vortex will pass through the top surface of the downstream cylinder, instead of attacking the front surface of the downstream cylinder. It is found 7

29 that the vortex shedding for V r <8 is similar to that shown in Fig. 7 (c) and that for V r >8 is similar to that shown in Fig. 7 (a). C L,rms CL CL CL+CL (a) (b) (c) (d) tf n Fig. 6 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and Vr= CL 5. VIV of two cylinders in the side-by-side arrangement Flow past two stationary cylinders in the side-by-side arrangement at Re=5 is firstly simulated and the calculated mean drag and the RMS lift coefficients are shown in Fig. 8. The two cylinders are labelled by Cylinders and, respectively as shown in Fig.. The statistic values of the force coefficients of Cylinder and those of Cylinder are the same because the symmetric configuration. The mean drag coefficient is greater than that of a single cylinder due to the increase in the blockage effect. The mean lift coefficient is very small at L/D=.5 and is increased to its maximum at L/D=4. The effect of the interference between the two cylinders is still obvious at L/D=6 judging by the mean drag and the RMS lift coefficients. 8

30 C L,rms CL CL CL+CL (a) (b) (c) (d) - CL tf n Fig. 7 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and Vr=8 (a) Mean Drag coefficient.8 (b) RMS lift coefficient.5.6 Cylinder Cylinder Single cylinder C D.5 Cylinder Cylinder Single cylinder C L,rms L/D L/D Fig. 8 Variations of mean drag and root mean square lift coefficients with L/D for two cylinders in the side-by-side arrangement at Re=5 Fig. 9 shows the time histories of the displacement of the two cylinders for the side-by-side arrangement. Alam et al. [7] found that the flow in the wake of two side-by-side cylinders was unstable if the gap ratio is less than.. In this study, the flow for L/D=.5 is found to be unstable at very small and very large reduced velocities (as shown in Fig. 9 (a) and (d)). The amplitude responses of the cylinders are perfectly periodic as shown in Fig. 9 (b) and (c). For L/D=.5 the 9

31 vibration of the cylinders is extremely irregular as V r = and 3. The very irregular vibrations are also found at L/D= and high reduced velocities as shown in Fig. 9 (i) and (j). For L/D=, the vibration is found to be periodic as V 3 and irregular as V 4. It can be found that the response r r of the cylinders at V r =3.5 is regular sometimes and irregular sometimes. The vibration of the two cylinders for L/D=4 and 6 are always regular and periodic in the whole range of simulated reduced velocities.. (a) L/D=.5,Vr=.5.5 (b) L/D=.5,Vr= (c) L/D=.5,Vr=5.5.5 (d) L/D=.5,Vr= (e) L/D=.5,Vr= (f) L/D=.5,Vr= (g) L/D=,Vr= (h) L/D=,Vr= (i) L/D=,Vr= (j) L/D=,Vr= (k) L/D=4,Vr= (l) L/D=4,Vr= (m) L/D=6,Vr= (n) L/D=6,Vr= tfn tfn Fig. 9 Time histories of the cylinder displacement for the side-by-side arrangement 3

32 Fig. Contours of the real part of the wavelet transform of the vibration displacement in the timefrequency plane for two cylinders in the side-by-side arrangement For L/D=.5 and, the response amplitude at reduced velocities higher than 5 are found to be highly irregular. Both vibration amplitude and vibrations change with time. Morlet wavelet analysis, which can identify the evolution of both amplitude and frequency, is used to analyse the response of the cylinder system. In the time domain the complex Morlet wavelet is expressed as / 4 i t / t / ( t) ( e e ) e, where ω is the nondimensional frequency, which is taken to be 6 to satisfy the admissibility condition [46, 47]. The Fourier transform of Morlet wavelet is ˆ ( ) / 4 ( ) / e and the continuous wavelet transform (CWT) of a signal X(t) is defined as 3

33 W x ( t, a) / a X ( ) [( t ) / a] d where a is the time scale. The local amplitude spectrum is calculated by A ( t, a) W ( t, a) S, where S is the peak amplitude of the Fourier transform of the x x / sampled wavelet [48]. Fig. shows the contours of the real part of the wavelet transform of the vibration displacement in the time-frequency plane for two cylinders in the side-by-side arrangement. The variation of the frequency with time can be clearly identified in Fig.. Take Fig. (a) as an example, the amplitude of displacement at frequency of f/f n = become strong and weak intermittently. The component with a frequency of about.5-.7 during 5<tf n < and 3<tf n <37 and 8<tf n <87 and is very weak in other time. Based on the wavelet analysis, it is found that vibrations corresponding to Fig. are the combination of VIV and galloping. Galloping is a classical instability mechanism where a small transverse body motion creates an aerodynamic force that increases the motion [49]. It is characterized by low-frequency oscillations that increase in amplitude unbounded with fluid velocity [5]. Galloping is driven by a time-averaged fluid force which develops in phase with the structural velocity and has a frequency many times lower than the vortex shedding frequency [5]. In Fig. (a) and (b), the highest frequency varies between.5 and.5. The Strouhal numbers for two side-by-side stationary cylinders at L/D=.5 and are S t =.66 and.7, respectively. If the nondimensional vibration frequency follows the Strouhal law, it should be f / f V S. The highest frequencies in Fig. (a), (b) and (c) are believed to be the frequencies n r t lead by vortex shedding. They are slightly less than the frequencies obtained from the Strouhal law due to the effects of the vibration. The high frequency component can be also clearly seen in Fig. 9 (e), (f), (i) and (j). In each of the wavelet spectrum in Fig., the low frequency component which is between.5 and are also observed. This frequency component, which is much lower than the vortex shedding frequency, is believed to be the results of galloping. The galloping is explained by Fig. 7 and associated discussion about Fig. 7 later on. The variation of the vibration frequency with time can be clearly seen in Fig., resulting in very irregular vibrations. It appears that the high frequency component becomes weak and the low frequency component becomes strong with an 3

34 increase in reduced velocity. This means that the galloping becomes strong and the VIV becomes weak as reduced velocity increases. The galloping amplitude increases and the VIV amplitude decreases can also be seen in the time histories in Fig. 9. Take L/D= as an example, the response is dominated by the high frequency component at V r =3.5 and the low frequency component at V r =3 (Fig. 9 (h) and (j)). In Fig. 9 (d) the high amplitude components has been extremely weak..8 (a) Amplitude for L/D=.5 and Side-by-side, L/D=.5 Side-by-side, L/D= Single cylinder.6 A (b) Amplitude for L/D=4 and 6.8 Side-by-side, L/D=4 Side-by-side, L/D=6 Single cylinder.6 A V r Fig. Variation of the response amplitude with the reduced velocity for the side-by-side arrangement 33

35 Fig. shows the variation of the response amplitude with the reduced velocity for the side-byside arrangement. The maximum calculated reduced velocity for L/D=.5 and are 3 in order to identify the galloping regime. Fig. shows the FFT spectra of the displacement and the lift coefficient for two cylinders in the side-by-side arrangement. Fig. 3 shows the variation of the response frequency with the reduced velocity for the side-by-side arrangement. The frequencies shown in Fig. 3 are the peak frequencies of the amplitude spectra. For L/D=.5 and, because the vibrations at V r >5 are so irregular that no distinct peak frequencies can be identified in the FFT spectra, only the frequencies at V 5 are shown in Fig. 3 (a). The lock-in regimes for L/D=.5 r and are found to be about 3.5 V r 5, which are much wider than that of a single cylinder. The response amplitude for L/D=.5 in the lock-in regime increases very slowly with the increasing reduced velocity and peaks at about V r =8.5. The maximum amplitude in the lock-in regime for L/D=.5 is almost, which is about 7% higher than that of a single cylinder. For L/D=.5 and, the response amplitude increases almost monotonically with the increasing reduced velocity as V r >5 until the maximum simulated reduced velocity of 3. In the study of galloping of a square cylinder Nemes et al. [5] found that the response frequency does not change with the reduced velocity and is close to the natural frequency of the cylinder. Although the vibration frequency at V r >5 cannot be identified due to the strong irregularity of the vibration, it can be seen in the wavelet spectra in Fig. that the nondimensional galloping frequency is generally less than and much smaller than the vortex shedding frequency. 34

36 A y /D.. (a) L/D=.5, V r =3.5 Y CL.5.5 A CL Ay/D.5 (b) L/D=.5, V r=5 Y CL.5.5 ACL A y /D.5 (c) L/D=.5, V r =8.5 Y CL.5.5 A CL A y /D 3 f/f n (d) L/D=.5, V r =.5.6 Y CL.4. A CL A y /D 3 f/f n. (e) L/D=.5, V r=8. Y CL.. A CL A y /D 3 f/f n.8 (f) L/D=, V r = Y CL. A CL Ay/D 3 3 f/f n f/f n.6 (g) L/D=, V r =.3. (h) L/D=, V r =5. Y Y.4 CL. CL... A CL A y /D A CL A y /D 3 f/f n. (a) L/D=, V r =5. Y CL.. ACL f/f n f/f n f/f n Fig. FFT spectra of the displacement and the lift coefficient for two cylinders in the side-by-side arrangement For two cylinders in the side-by-side arrangement at L/D=4 and 6, the response amplitude and frequency have been very similar to that of a single cylinder. The lock-in regime becomes closer to that of a single cylinder as the gap ratio L/D is increased from 4 to 6. Different from that of a single cylinder, the response amplitude in the range of V 3 is exactly zero for L/D=4 and 6. The amplitude is exactly zero because the flow is exactly symmetric as discussed later on. Outside the lock regime the response frequency increases linearly with the reduced velocity, indicating that it follows the Strouhal law. Based on the response amplitude and frequency it can be seen that for all the gap ratios, the lower boundaries of the lock-in regime are the same and they are 3.5. r 35

37 3.5 (a) Frequency for L/D=.5 and Side-by-side, L/D=.5 Side-by-side, L/D= Single cylinder f/f n (b) Frequency for L/D=4 and 6 Side-by-side, L/D=4 Side-by-side, L/D=6 Single cylinder f/f n Fig. 3 Variation of the response frequency with the reduced velocity for the side-by-side arrangement V r Fig. 4 shows the variation of the RMS lift coefficient on the two cylinders in the tandem arrangement with the reduced velocity. The RMS lift coefficient starts to increases with the increasing reduced velocity at about V r =3.5 for all the gap ratios, which is the lower boundary of the lock-in regime. For a single cylinder, after the RMS lift coefficient peaks at V r =3.5, it reduces very quickly to a very small value at V r =5.5. It can be seen that both the RMS lift coefficient for each individual cylinder and the total RMS lift coefficient are significantly greater than that of a single cylinder in the reduced velocity range of 5.5 V r 5, resulting in much wider lock-in regime than that of a single cylinder. As V r >5, the RMS lift coefficient is almost independent on the reduced 36

38 velocity for L/D=.5 and. The RMS lift coefficient of the total force for L/D=.5 as 3.5 V r 5 is slightly greater than that for L/D=. For L/D=4 and 6, the variations of the RMS lift coefficient with the reduced velocity is very similar to that of a single cylinder. The RMS lift coefficient of the total force as V 3 is zero because the vortex shedding flow is exactly symmetric as shown in Fig. 3. r.5 (a) C L,rms of Cylinder for L/D=.5 and (b) C L,rms of Cylinder for L/D=4 and 6 CL,rms.5 Side-by-side L/D=.5 Side-by-side L/D= Single cylinder CL,rms.5 Side-by-side L/D=4 Side-by-side L/D=6 Single cylinder (c) C L,rms of Cylinder for L/D=.5 and (d) C L,rms of Cylinder for L/D=4 and 6 CL,rms.5 Side-by-side L/D=.5 Side-by-side L/D= Single cylinder CL,rms.5 Side-by-side L/D=4 Side-by-side L/D=6 Single cylinder (e) C L,rms of total force for L/D=.5 and (f) C L,rms of total force for L/D=4 and 6 CL,rms.5 Side-by-side L/D=.5 Side-by-side L/D= Single cylinder CL,rms Side-by-side L/D=4 Side-by-side L/D=6 Single cylinder Vr Fig. 4 RMS lift coefficient on the two cylinders in the side-by-side arrangement Vr Fig. 5 shows the variation of the phase ψ between the response displacement and the total lift force with the reduced velocity for two cylinders in the side-by-side arrangement. Similar to that of a single cylinder, the phase ψ jumps from to 8 at a reduced velocity for each gap ratio. The phase changes from to 8 at V r = 8.5 and, respectively for L/D=.5 and. At L/D=4 and 6 the phase changes happens at V r =5.5. The phase ψ for a gap ratio with narrower lock-in regime changes from to 8 at smaller reduced velocity than that for a gap ratio with wider lock-in 37

39 regime. The variations of phase with the reduced velocity at L/D=4 and 6 are very similar to each other due to the weak effects of the gap on the flow and VIV. 5 8 (a) Phase between Y and C L for L/D=.5 and 35 Side-by-side, L/D=.5 ψ (degree) 9 45 Side-by-side, L/D= (b) Phase between Y and C L for L/D=4 and 6 35 ψ (degree) 9 45 Side-by-side, L/D=4 Side-by-side, L/D= V r Fig. 5 Variation of the phase between the response displacement and the total lift force with the reduced velocity for two cylinders in the side-by-side arrangement Fig. 6 shows the vorticity contours for two cylinders in a side-by-side arrangement at L/D=.5 and V r =8.5, where the response amplitude in the cross-flow direction peaks. In Fig. 6 (a) and (b), the positive vortex from the bottom of the top cylinder is very weak and the two negative vortices from the tops of the two cylinders combine. The combined negative vortex starts to be shed from the cylinder in Fig. 6 (c), followed by the combination of the two positive vortices from the bottoms of the two cylinders. The vortex shedding from the bottom of the cylinders when the cylinders are at their lowest position makes a negative lift force on the cylinder system. This makes the lift force in- 38

40 phase with the response displacement. In each vibration cycle, two main vortices are shed from the cylinder and the vortex street in the wake of the two cylinders is very similar to that of a single cylinder. The two cylinders behave like a single body. The lift coefficients of the two cylinders shown at the bottom of Fig. 6 are more or less in anti-phase with each other, leading to a total lift coefficient with amplitude smaller than that of either of the cylinders. The lift coefficient of the total force is in phase with the vibration displacement. In Fig. 6 (d), very small portion of the positive vortex generated from the bottom of the top cylinder is cut off from its main part. This vortex is too weak to have effect on the VIV. C L,rms CL CL CL+CL (a)(b) (c) (d) (e) (f) tf n C L Fig. 6 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=.5 and Vr=8.5 39

41 Fig. 7 shows the vorticity contours for two cylinders in a side-by-side arrangement at L/D=.5 and V r =5, where the galloping occurs. The vortices from the gap between the cylinders are stronger than those in Fig. 6, making it harder for the two vortices of the same direction from the two cylinders to combine with each other. It can be seen in Fig. 7 (a) and (b) that after the negativee vortex is shed from the top surface of the top cylinder, it is connected with the negative vortex from the top surface of the bottom cylinder, which has been separated from the cylinder surface. These two negative vortices are separated from each other in Fig. 7 (c), due to the combination effects of the two positive vortices from the bottoms of the cylinders. The vortex shedding flow in one vibration period is found different from that in another vibration period. It is found that whenever the galloping occurs (Fig. 7 (a) (g)), the vortices from the gap between the two cylinders are strong. As the response amplitude is increased from its lowest position in Fig. 7 (c) to its highest position in Fig. 7 (g), the lift coefficients on both cylinders oscillate about three periods. Apart from the vortex shedding, it is found that the change of the direction of the jet flow from the gap between the cylinders also contributes the oscillation of the lift coefficient. For example, the jet flow between the gap in Fig. 7 (c) biases upwards and that in Fig. 7 (d) biases downwards. Although there is not obvious vortex shedding, the lift coefficients of both cylinder oscillates almost one period between Fig. 7 (c) and (d). The switching of the gap flow from one side to the other is known as flipflopping flow [5, 6, 4]. The vortex shedding flow in Fig. 7 (h), (i) and (j), which is within the regular VIV period between tf n =53.7 to 54.7, is very similar to that in Fig. 6. After tf n =55, galloping occurs again with the response displacement continuously increasing for a long period of time. 4

42 (a)(b)(c)(d)(e)(f)(g) CL CL CL+CL (h) (i) (j).5 CL,rms C L tf n Fig. 7 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=.5 and V r =5 4

43 Fig. 8 shows the vorticity contours for two cylinders in a side-by-side arrangement at L/D= and V r =4.5, where the response amplitude peaks. Compared with those for L/D=.5, vortex shedding flows from the two cylinders are more independent on each other. The negative vortices from the tops of the two cylinders are shed from the cylinders separately without combination as shown in Fig. 8 (a), (b) and (c). The two positive vortices from the bottoms of the cylinders are also shed from the cylinders separately. They combined after they are shed from the cylinders. The vortices interact with each other after they are shed from the cylinders, resulting very irregular vortex street. C L,rms CL CL CL+CL (a) (b) (c) (d) (e) (f) C L tf n Fig. 8 Vorticity contours for two cylinders in a side-by-side arrangement at L/D= and Vr=4.5 4

44 Fig. 9 shows the vorticity contours for two cylinders in a side-by-side arrangement at L/D=4 and V r =6. The interaction between the two cylinders has been very weak at L/D=4 and the vortex shedding flows from the two cylinders are in phase with each other. In each vibration period, two vortices (a pair) are shed from each cylinder. The interaction between the two vortices of the same direction from the two cylinder may occurs, but in the far downstream area of the cylinders. The vortex shedding from each cylinder is very similar to that of a single cylinder and the vibration frequency is the same as the vortex shedding frequency. The vortex shedding from each cylinder is in S mode..5 CL CL CL+CL (a) (b) (c) (d) C L,rms CL tf n Fig. 9 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=4 and Vr=6 Fig. 3 shows the vorticity contours for two cylinders in a side-by-side arrangement at L/D=6, V r =3 and L/D=4, V r =3, where the response amplitude is zero. The response amplitude is zero because the vortex shedding from one cylinder is wholly in anti-phase with that from another cylinder. The anti-phase between the two vortex streets leads zero lift coefficient and in turn no vibration of the 43

45 cylinders. In order to testify if the anti-phase between the two vortex streets is intrinsic, separate simulations are conducted for V r =.5 to 3 at L/D=4 and 6 with an initial displacement of the cylinder in the cross-flow direction. Fig. 3 (c) is an example of the time history of the vibration displacement of the cylinders for L/D=6 and V r =3. It can be seen that the initial displacement of the cylinders decreases gradually with time and finally become zero. Fig. 3 (c) demonstrates that the flow pattern in the wake of the two cylinders is intrinsically symmetric. Even an asymmetric initial condition will leads to symmetric flow pattern. (c) Time histories for L/D=6, V r =3. Initial disturbation is introduced tf n Fig. 3 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=6, Vr=3 and L/D=4, Vr=3 6. Conclusions Flow-induced vibration of two cylinders in tandem and side-by-side arrangements at a low Reynolds number of 5 are studied numerically by solving the D Navier-Stokes equations using FEM. The numerical model is carefully validated against the published data of flow past two fixed cylinders, VIV of a single cylinder and VIV of one elastically mounted cylinder in the wake of the fixed cylinder. The conclusions are summarized as follows. 44

46 6. Tandem arrangement When the two cylinders are in the tandem arrangement, the critical gap for vortex shedding from the upstream cylinder is significantly less than that for two stationary cylinders. If the two cylinders are stationary, the vortex shedding from the upstream cylinder does not occur until the gap ratio L/D exceeds 3.5 or 4. However, if the two cylinders are elastically mounted, the vortex shedding from the upstream cylinder happens at L/D= in the lock-in regime. In the tandem arrangement, the lock-in regime of the reduced velocity is narrower than that of a single cylinder for L/D=.5 and and wider than later for L/D=4 and 6. The response amplitude at L/D=.5 is much smaller than that of a single cylinder. For L/D=4 and 6, the maximum response amplitude in the lock-in regime is much greater than that of a single cylinder. For L/D=6 and in the range of 9 V 3. r 5, the lift force has two frequency components and because secondary lift component locks on the natural frequency, the response amplitude in the lock-in regime is small. The vortex shedding from the upstream cylinder has significant effects on the response. The vortices that are shed from the upstream cylinder may scratch the top and the bottom surface or impinge the front surface of the downstream cylinder, depending on the reduced velocity. 6. Side-by-side arrangement The most interesting phenomenon found in the side-by-side arrangement is the combination of VIV and galloping at L/D=.5 and. For L/D=.5 and, the response is dominated by VIV as V r <5 and by galloping as V r >5. At reduced velocities close to 5, the response is a combination of VIV and galloping. The Wavelet transform is used to identify that the vibration frequencies at reduced velocities greater than 5 include the vortex shedding frequency and the galloping frequency. With an increase in the reduced velocity, the amplitude corresponding to VIV becomes weak and that 45

47 corresponding to the galloping becomes strong. Due to galloping the response amplitude increases continuously with the increasing reduced velocity as V r >5. The response of the two cylinder at large L/D=4 and 6 is very similar to that of a single cylinder due to the weak interaction between the two vortex streets. The vortex shedding flows at these two L/D are in-phase with each other in the lock-in regime. They are in anti-phase with each other as V 3 even with an asymmetric initial condition, resulting in absolutely no vibration. r The vortex shedding from the two cylinders for L/D=.5 and is very similar to that of a single cylinder in the VIV regime. During the vortex shedding, the two negative vortices from the cylinders combine and the two positive vortices combine, leaving two rows of vortices in the wake of the cylinders. When the response is in the galloping regime, apart from the vortex shedding, the change in the jet flow direction also leads to the oscillatory of the lift force. The galloping frequency is much smaller than the frequency of the lift force. References. C.H.K.Williamson and A. Roshko, Vortex formation in the wake of an oscillating cylinder, Journal of Fluids and Structures, (988).. T. Sarpkaya, A critical review of the intrinsic nature of vortex-induced vibrations, Journal of Fluids and Structures 9 (4), (4). 3. C.H.K Williamson and R. Govardhan, Vortex-induced vibrations, Annual Review of Fluid Mechanics 36, (4). 4. A. Khalak, and C.H.K Williamson, Motions, force and mode transitions in vortex-induced vibrations at low mass-damping, Journal of Fluids and Structures 3, (999). 5. N. Jauvtis and C.H.K Williamson, The effect of two degrees of freedom on vortex-induced vibration at low mass and damping, Journal of Fluid Mechanics 59, 3-6 (4). 6. R. Govardhan and C.H.K Williamson, Modes of vortex formation and frequency response of a freely vibrating cylinder, Journal of Fluid Mechanics 4, 85-3 (). 7. S.P. Singh and S. Mittal, S., 5, Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes, Journal of Fluids and Structures, 85 4 (5). 46

48 8. M. Zhao, L. Cheng, and T. Zhou, T., Numerical simulation of vortex-induced vibration of a square cylinder at a low Reynolds number, Physics of Fluids 5, 363 (3). 9. C.C.Feng, C. C., The measurements of vortex-induced effects on flow past a stationary and oscillation and galloping, Master s thesis, University BC, Vancouver, Canada (968).. P.W. Bearman, Vortex shedding from oscillating bluff_bodies, Annual Review of Fluid Mechanics 6, 95- (984).. M.M. Zdravkovich, Review of flow interference between two circular cylinders in various arrangements, ASME Journal of Fluids Engineering 99, (977).. M.M. Zdravkovich, Flow induced oscillations of two interfering circular cylinders, Journal of Sound and Vibration, 5-5 (985). 3. M.M. Zdravkovich, The effects of interference between circular cylinders in cross flow, Journal of Fluids and Structures, pp (987). 4. C.H.K. Williamson, Evolution of a single wake behind a pair of bluff bodies, Journal of Fluid Mechanics 59, -8 (985). 5. P.W. Bearman and A.J. Wadcock, The interaction between a pair of circular cylinders normal to a stream, Journal of Fluid Mechanics 6, (973). 6. H.J. Kim and P.A. Durbin, Investigation of the flow between a pair of cylinders in the flopping regime, Journal of Fluid Mechanics, 96, (988). 7. M.M. Alam, M. Moriya, and H. Sakamoto, Aerodynamic characteristics of two side-by-side circular cylinders and application of wavelet analysis on the switching phenomenon, Journal of Fluids and Structures 8, (3). 8. J.R. Meneghini, F. Saltara, C.L.R. Siqueira, and J.J.A Ferrari, Numerical simulation of flow interference between two circular cylinders in tandem and side-by-side arrangements, Journal of Fluids and Structures 5 (), (). 9. J. Mizushima and N. Suehiro, Instability and transition of flow past two tandem circular cylinders, Physics of Fluids 7 (), 47 (5).. Y. Tasaka, S. Kon, L. Schouveiler, and P.L. Gal, Hysteretic mode exchange in the wake of two circular cylinders in tandem, Physics of Fluids 8, 844 (6).. M.M. Zdravkovich, Review of interference-induced oscillations in flow past two parallel circular cylinders in various arrangements, J. Wind Engineering and Industrial Aerodynamics 8, 83 (988). E. Fontaine, J. Morel, Y. Scolan, and T. Rippol, Riser interference and VIV amplification in tandem configuration, International Journal of Offshore and Polar Engineering 6 (), 33 4 (6). 47

49 3. H.P. Ruscheweyh, Aeroelastic interference effects between slender structures, Journal of Wind Engineering and Industrial Aerodynamics. 4 ( 3), 9-4 (983). 4. D. Brika, and A. Laneville, The flow interaction between a stationary cylinder and a downstream flexible cylinder, Journal of Fluids and Structures 3 (5), (999). 5. A. Bokaian and F. Geoola, Wake-induced galloping of two interfering circular cylinders, Journal of Fluid Mechanics 46, (984). 6. S. Mittal and V. Kumar, Flow-induced oscillations of two cylinders in tandem and staggered arrangements, Journal of Fluids and Structures 5 (5), (). 7. S. Mittal and V. Kumar, V., Vortex induced vibrations of a pair of cylinders at Reynolds number, International Journal of Computational Fluid Dynamics 8, 6-64 (4). 8. W. Jester and Y. Kallinderis, Numerical study of incompressible flow about transversely oscillating cylinder pairs, Journal of Offshore Mechanics and Arctic Engineering 6 (4), 3-37 (4). 9. Z. Zang, F. Gao, and J.S. Cui, 3, Physical modeling and swirling strength analysis of vortex shedding from near-bed piggyback pipelines, Applied Ocean Research 4, 5-59 (3). 3. M. Zhao and G. Yan, Numerical simulation of vortex-induced vibration of two circular cylinders of different diameters at low Reynolds number, Physics of Fluids 5, 836 (3). 3. F.J. Huera-Huarte, and M. Gharib, M., Flow-induced vibrations of a side-by-side arrangement of two flexible circular cylinders, Journal of Fluids and Structures 7, (). 3. X.Q. Wang, R.M.C. So, W.C. Xie, and J. Zhu, J., Free-stream turbulence effects on vortexinduced vibration of two side-by-side elastic cylinders, Journal of Fluids and Structures 4, (8). 33. C.H.K. Williamson, The existence of two stages in the transition to three-dimensionality of a cylinder wake, Physics of Fluids 3, (988). 34. C.H.K. Williamson, Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers, Journal of Fluid Mechanics 6, (989). 35. I. Borazjani and F. Sotiropoulos, Vortex-induced vibrations of two cylinders in tandem arrangement in the proximity-wake interference region, Journal of Fluid Mechanics 6, (9). 36. M. Zhao and L. Cheng, Numerical simulation of two-degree-of-freedom vortex-induced vibration of a circular cylinder close to a plane boundary, Journal of Fluids and Structures 7, 97 (). 37. M. Zhao and L. Cheng, Numerical investigation of local scour below a vibrating pipeline under steady currents, Coastal Engineering 57, (). 48

50 38. M. Zhao, F. Tong, and L. Cheng, Numerical simulation of two-degree-of-freedom vortexinduced vibration of a circular cylinder between two lateral plane walls in steady currents, Journal of Fluids Engineering, 34, 45 (). 39. Y. Bao, D. Zhou, and J. Tu, Flow interference between a stationary cylinder and an elastically mounted cylinder arranged in proximity, Journal of Fluids and Structures 7, (). 4. H.T. Ahn and Y. Kallinderis, Strongly coupled flow/structure interactions with a geometrically conservative ALE scheme on general hybrid meshes, Journal of Computational Physics 9, (6). 4. G.R.S. Assi, J.R. Meneghini, J.A.P. Aranhaa, P.W. Bearman, and E. Casaprima, Experimental investigation of flow-induced vibration interference between two circular cylinders, Journal of Fluids and Structures, (6). 4. M.M. Zdravkovich and D.L. Pridden, Interference between two circular cylinders, series of unexpected discontinuities, Journal of Industrial Aerodynamics,, 55-7 (977). 43. T.K., Prasanth, S. Sehara, S.P. Singh, R. Kumar, and S. Mittal, Effect of blockage on vortexinduced vibrations at low Reynolds numbers, Journal of Fluids and Structures, (6). 44. B.M. Sumer and J. Fredsøe, Hydrodynamics around Cylindrical Structures. World Scientific (997). 45. S. Mittal and V. Kumar, Finite element study of vortex-induced cross-flow and in-line oscillations of a circular cylinder at low Reynolds numbers, International Journal for Numerical Methods in Fluids 3, 87- (999). 46. M. Farge, Wavelet transforms and their applications to turbulence, Annual Review of Fluid Mechanics 4, (99). 47. R.N., Leao and J.A. Burne, J.A., Continuous wavelet transform in the evaluation of stretch reflex response from surface EMG, Journal of Neuroscience Methods 33, 5-5 (4). 48. B. Nener, C. Ridsdill-Smith, and T.A. Zeisse, Wavelet analysis of flow altitude infrared transmission in the coastal environment, Infrared Physics & Technology 4, (999). 49. P.W. Bearman, I.S Gartshore, D.J. Maull, and G.V. Parkinson, Experiments on flow-induced vibration of a square-section cylinder, Journal of Fluids and Structures, 9-34 (987). 5. A. Nemes, J. Zhao, D.L. Jacono, and J. Sheridan, The interaction between flow-induced vibration mechanisms of a square cylinder with varying angles of attack, Journal of Fluid Mechanics 7, -3 (). 49

51 5. I. Robertson, L. Li, S.J. Sherwin, and P.W Bearman, P.W., A numerical study of rotational and transverse galloping rectangular bodies, Journal of Fluids and Structures 7, (3). 5

52 List of Figure captions Fig. Sketch of flow past two cylinders in (a) tandem and (b) side-by-side arrangements Fig. Computational mesh near the two cylinders Fig. 3 Variations of the response amplitude and the force coefficients with the reduced velocity for a single cylinder at Re=5, M red = and ζ=. Fig. 4 The response amplitude and the root mean square lift coefficient of an elastically mounted cylinder in the wake of stationary cylinder with L/D=4, M red = and ζ= Fig. 5 Variations of the mean drag and root mean square lift coefficients with L/D for two stationary cylinders in the tandem arrangement Fig. 6 Time histories of the vibration displacement for the tandem arrangement Fig. 7 Amplitude spectra of the vibration based on the FFT analysis Fig. 8 Variation of the response amplitude with the reduced velocity for the tandem arrangement Fig. 9 Variation of the response frequency with the reduced velocity for the tandem arrangement Fig. Variation of the RMS lift coefficient with the reduced velocity for two cylinders in the tandem arrangement Fig. Variation of the phase between the response displacement and the total lift force with the reduced velocity for tandem arrangement Fig. Vorticity contours around two cylinders in a tandem arrangement at L/D=.5 and V r =4.5 Fig. 3 Vorticity contours for two cylinders in a tandem arrangement at L/D= and V r =5 Fig. 4 Vorticity contours for two cylinders in a tandem arrangement at L/D=4 and V r =6 Fig. 5 Vorticity contours for two cylinders in a tandem arrangement when the cylinders are at the lowest location for L/D=4 and V r =4, 4.5, 5 and 7 Fig. 6 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and V r =4 Fig. 7 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and V r =8 Fig. 8 Variations of mean drag and root mean square lift coefficients with L/D for two cylinders in the side-by-side arrangement at Re=5 Fig. 9 Time histories of the cylinder displacement for the side-by-side arrangement Fig. Contours of the real part of the wavelet transform of the vibration displacement in the timefrequency plane for two cylinders in the side-by-side arrangement Fig. Variation of the response amplitude with the reduced velocity for the side-by-side arrangement Fig. FFT spectra of the displacement and the lift coefficient for two cylinders in the side-by-side arrangement 5

53 Fig. 3 Variation of the response frequency with the reduced velocity for the side-by-side arrangement Fig. 4 RMS lift coefficient on the two cylinders in the side-by-side arrangement Fig. 5 Variation of the phase between the response displacement and the total lift force with the reduced velocity for two cylinders in the side-by-side arrangement Fig. 6 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=.5 and V r =8.5 Fig. 7 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=.5 and V r =5 Fig. 8 Vorticity contours for two cylinders in a side-by-side arrangement at L/D= and V r =4.5 Fig. 9 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=4 and V r =6 Fig. 3 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=6, V r =3 and L/D=4, V r =3 5

54 Table Comparison of the numerical results of flow past two stationary cylinders in tandem Reference L/D C D A CL S t C D A CL S t Borazjani and Sotiropoulos (9) Meneghini et al. () Present Meneghini et al. () Present Borazjani and Sotiropoulos (9) Meneghini et al. () Present

55 (a) y L (b) y D Cylinder Cylinder Cylinder Flow D O D x Flow O L x D Cylinder Fig. Sketch of flow past two cylinders in (a) tandem and (b) side-by-side arrangements

56 Fig. Computational mesh near the cylinders

57 .8 (a) Responseamplitude Ahn and Kallinderis [4] Borazjani and Sotiropoulos [35] Bao et al. [39] Present.6 A y /D V r Force coefficient (b) Force coefficient Mean CD, Bao et al. [39] Mean CD, Present RMS CL, Bao et al. [39] RMS CL, Present RMS CD, Bao et al. [39] RMS CD, Present V r Fig. 3Variations of the response amplitude and the force coefficients with the reduced velocity for a single cylinder at Re=5, M red = and ζ=.

58 .4. (a) Response amplitude of the downstream cylinder.5 (b) Root mean square of the lift coefficient of the downstream cylinder Ay/D Bao et al. [39] Present Assi et al. [4] m *, m * ζ 3, Re=3-3 C L,rms.5 Bao et al. [39] Present V r V r Fig. 4 The response amplitude and the root mean square lift coefficient of an elastically mounted cylinder in the wake of stationary cylinder with L/D=4, M red = and ζ=

59 3.5 (a) Mean Drag coefficient Upstream cylinder Downstream cylinder Single cylinder Exp., downstream cylinder, Re=3 [4].5 (b) RMS lift coefficient Upstream cylinder Downstream cylinder Single cylinder CD.5 C L,rms L/D L/D Fig. 5 Variations of the mean drag and root mean square lift coefficients with L/D for two stationary cylinders in the tandem arrangement

60 .4. (a) L/D=.5,Vr= (b) L/D=.5,Vr= (c) L/D=,Vr= (d) L/D=,Vr= (e) L/D=,Vr= (f) L/D=,Vr= (g) L/D=4,Vr= (h) L/D=4,Vr= (i) L/D=6,Vr= (j) L/D=6,Vr= (k) L/D=6,Vr= (l) L/D=6,Vr= tfn tfn Fig. 6 Time histories of the cylinder displacement for the tandem arrangement

61 . (a) L/D=.5, Vr= (b) L/D=.5, V r =5.8.4 (c) L/D=, V r =5.5 Ay/D. Y CL.6.4. A CL A y /D. Y CL.6.4. A CL Ay/D.3.. Y CL.5 A CL 3 f/f n. (d) L/D=, V r =6.3 3 f/fn. (e) L/D=6, V r =9.4 3 f/f n. (f) L/D=6, V r =.4 Ay/D. Y CL.. A CL A y /D. Y CL.3.. A CL Ay/D. Y CL.3.. A CL 3 f/f n 3 f/f n 3 f/f n Fig. 7 Amplitude spectra of the vibration based on the FFT analysis

62 .8 (a) Amplitude for L/D=.5 and Tandem, L/D=.5 Tandem, L/D=.6 Single cylinder Ay/D (b) Amplitude for L/D=4 and 6 Tandem, L/D=4 Tandem, L/D=6.6 Single cylinder Ay/D V r Fig. 8 Variation of the response amplitude with the reduced velocity for the tandem arrangement

63 3.5 (a) Frequency for L/D=.5 and Tandem, L/D=.5 Tandem, L/D= Single cylinder f/fn.5 f/f n = (b) Frequency for L/D=4 and 6 Tandem, L/D=4 Tandem, L/D=6 f/fn.5 Single cylinder f/f n = V r Fig. 9 Variation of the response frequency with the reduced velocity for the tandem arrangement

64 (a) C L,rms of the upstream cylinder CL,rms.5 Tandem, L/D=.5 Tandem, L/D= Tandem, L/D=4 Tandem, L/D=6.5 C L,rms (b) C L,rms of the downstream cylinder Tandem, L/D=.5 Tandem, L/D= Tandem, L/D=4 Tandem, L/D= (c) C L,rms of the total force.5 Tandem, L/D=.5 Tandem, L/D= Tandem, L/D=4 Tandem, L/D=6 CL,rms V r Fig. Variation of the RMS lift coefficient with the reduced velocity for two cylinders in the tandem arrangement

65 5 (c) Phase between Y and C L for L/D=.5 and 8 35 Tandem, L/D=.5 ψ (degree) 9 Tandem, L/D= (c) Phase between Y and C L for L/D=4 and 6 35 Tandem, L/D=4 Tandem, L/D=6 ψ (degree) V r Fig. Variation of the phase between the response displacement and the total lift force with the reduced velocity for tandem arrangement

66 CL,rms CL CL CL+CL (a) (b) (c) (d) CL tf n Fig. Vorticity contours for two cylinders in a tandem arrangement at L/D=.5 and V r =4.5

67 C L,rms CL CL CL+CL 4 (a) (b) (c) (d) (e) (f) tf n CL Fig. 3 Vorticity contours for two cylinders in a tandem arrangement at L/D= and V r =5

68

69 CL,rms CL CL CL+CL (a) (b) (c) (d) - CL tf n Fig. 4 Vorticity contours for two cylinders in a tandem arrangement at L/D=4 and V r =6

70 Fig. 5 Vorticity contours for two cylinders in a tandem arrangement when the cylinders are at the lowest location at L/D=4 and V r =4, 4.5, 5 and 7

71 C L,rms CL CL CL+CL (a) (b) (c) (d) tf n CL Fig. 6 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and V r =4

72 CL,rms CL CL CL+CL (a) (b) (c) (d) - C L tf n Fig. 7 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and V r =8

73 (a) Mean Drag coefficient.8 (b) RMS lift coefficient.5.6 Cylinder Cylinder Single cylinder C D.5 Cylinder Cylinder Single cylinder C L,rms L/D L/D Fig. 8 Variations of mean drag and root mean square lift coefficients with L/D for two cylinders in the side-by-side arrangement at Re=5

74 . (a) L/D=.5,Vr=.5.5 (b) L/D=.5,Vr= (c) L/D=.5,Vr=5.5.5 (d) L/D=.5,Vr= (e) L/D=.5,Vr= (f) L/D=.5,Vr= (g) L/D=,Vr= (h) L/D=,Vr= (i) L/D=,Vr= (j) L/D=,Vr= (k) L/D=4,Vr= (l) L/D=4,Vr= (m) L/D=6,Vr= (n) L/D=6,Vr= tfn tfn Fig. 9 Time histories of the cylinder displacement for the side-by-side arrangement

75 Fig. Contours of the real part of the wavelet transform of the vibration displacement in the timefrequency plane for two cylinders in the side-by-side arrangement

76 .8 (a) Amplitude for L/D=.5 and Side-by-side, L/D=.5 Side-by-side, L/D= Single cylinder.6 A (b) Amplitude for L/D=4 and 6.8 Side-by-side, L/D=4 Side-by-side, L/D=6 Single cylinder.6 A V r Fig. Variation of the response amplitude with the reduced velocity for the side-by-side arrangement

77 A y /D.. (a) L/D=.5, V r =3.5 Y CL.5.5 A CL Ay/D.5 (b) L/D=.5, V r=5 Y CL.5.5 ACL A y /D.5 (c) L/D=.5, V r =8.5 Y CL.5.5 A CL A y /D 3 f/f n (d) L/D=.5, V r =.5.6 Y CL.4. A CL A y /D 3 f/f n. (e) L/D=.5, V r=8. Y CL.. A CL A y /D 3 f/f n.8 (f) L/D=, V r = Y CL. A CL Ay/D 3 3 f/f n f/f n.6 (g) L/D=, V r =.3. (h) L/D=, V r =5. Y Y.4 CL. CL... A CL A y /D A CL A y /D 3 f/f n. (a) L/D=, V r =5. Y CL.. ACL 3 f/f n 3 4 f/f n 3 4 f/f n Fig. FFT spectra of the displacement and the lift coefficient for two cylinders in the side-by-side arrangement

78 3.5 (a) Frequency for L/D=.5 and Side-by-side, L/D=.5 Side-by-side, L/D= Single cylinder f/f n (b) Frequency for L/D=4 and 6 Side-by-side, L/D=4 Side-by-side, L/D=6 Single cylinder f/f n V r Fig. 3 Variation of the response frequency with the reduced velocity for the side-by-side arrangement

79 .5 (a) C L,rms of Cylinder for L/D=.5 and (b) C L,rms of Cylinder for L/D=4 and 6 CL,rms.5 Side-by-side L/D=.5 Side-by-side L/D= Single cylinder CL,rms.5 Side-by-side L/D=4 Side-by-side L/D=6 Single cylinder (c) C L,rms of Cylinder for L/D=.5 and (d) C L,rms of Cylinder for L/D=4 and 6 CL,rms.5 Side-by-side L/D=.5 Side-by-side L/D= Single cylinder CL,rms.5 Side-by-side L/D=4 Side-by-side L/D=6 Single cylinder (e) C L,rms of total force for L/D=.5 and (f) C L,rms of total force for L/D=4 and 6 CL,rms.5 Side-by-side L/D=.5 Side-by-side L/D= Single cylinder CL,rms Side-by-side L/D=4 Side-by-side L/D=6 Single cylinder Vr Vr Fig. 4 RMS lift coefficient on the two cylinders in the side-by-side arrangement

80 5 (a) Phase between Y and C L for L/D=.5 and 8 35 Side-by-side, L/D=.5 ψ (degree) 9 Side-by-side, L/D= (b) Phase between Y and C L for L/D=4 and 6 35 ψ (degree) 9 45 Side-by-side, L/D=4 Side-by-side, L/D= V r Fig. 5 Variation of the phase between the response displacement and the total lift force with the reduced velocity for two cylinders in the side-by-side arrangement

81 CL,rms CL CL CL+CL (a)(b) (c) (d) (e) (f) tf n CL Fig. 6 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=.5 and V r =8.5

82 (a)(b)(c)(d)(e)(f)(g) CL CL CL+CL (h) (i) (j).5 CL,rms C L tf n Fig. 7 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=.5 and V r =5.

83 C L,rms CL CL CL+CL (a) (b) (c) (d) (e) (f) C L tf n Fig. 8 Vorticity contours for two cylinders in a side-by-side arrangement at L/D= and V r =4.5

84 .5 CL CL CL+CL (a) (b) (c) (d) C L,rms CL tf n Fig. 9 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=4 and V r =6