numerical simulation of flow over two side-by-side circular cylinders

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2011,23(6):792-805 DOI: 10.1016/S1001-6058(10)60178-3

NUMERICAL SIMULATION OF FLOW OVER TWO SIDE-BY-SIDE CIR- CULAR CYLINDERS*

SARVGHAD-MOGHADDAM Hesam, NOOREDIN Navid Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran, E-mail: [email protected] GHADIRI-DEHKORDI Behzad Department of Mechanical Engineering, School of Engineering, Tarbiat Modares University, Tehran, Iran

(Received March 21, 2011, Revised May 20, 2011)

Abstract: In the present paper, the unsteady, viscous, incompressible and 2-D flow around two side-by-side circular cylinders was simulated using a Cartesian-staggered grid finite volume based method. A great-source term technique was employed to identify thesolid bodies (cylinders) located in the flow field and boundary conditions were enforced by applying the ghost-cell technique. Finally, the characteristics of the flow around two side-by-side cylinders were comprehensively obtained through several computational simulations. The computational simulations were performed for different transverse gap ratios ( 1.5 / 4T D≤ ≤ ) in laminar ( = 100, 200Re ) and turbulent ( 4= 10Re ) regimes, where T and D are the distance between the centers of cylinders and the diameter of cylinders, respectively. The Reynolds number is based on the diameter of cylinders, D . The pressure field and vorticity distributions along with the associated streamlines and the time histories of hydrodynamic forces were also calculated and analyzed for different gap ratios. Generally, different flow patterns were observed as the gap ratio and Reynolds number varied. Accordingly, the hydrodynamic forces showed irregular variations for small gaps while they took a regular pattern at higher spacing ratios.

Key words: side-by-side cylinders, vortex shedding, flow induced forces, finite volume method, turbulent flow

Introduction

Flow past a group of circular cylinders in different arrangements plays an important role in various engineering applications and it has been the subject of many researches for decades[1]. Among these arrangements, two-cylinder configurations have shown to be more effective in different areas such as flow past heat exchanger tubes, transmission cables, offshore plat- forms and chimney stacks. Furthermore, study of flow past two circular cylinders provides a broad perspec- tive toward understanding the complex behavior of different arrangements. With respect to cross-flow, the configuration of two circular cylinders is divided to tandem, side-by-side and staggered arrangements. This complexity arises due to interaction of shed vortices from the two cylinders in close proximity,

* Biography: SARVGHAD-MOGHADDAM Hesam (1984-), Male, Master

separation of shear layers from the surfaces of these cylinders and propagation of wakes behind these cylinders, especially in side-by-side arrangement[2].

Accordingly, various experimental and numerical studies have been carried out on flow past two side- by-side circular cylinders. Of experimental researches who have studied the unsteady, viscous and incompressible flow around this arrangement[1,3-5] .

However, in an experimental work on two side- by-side cylinders at Reynolds numbers ( )Re ranging from 50 to 200, Williamson[3] realized harmonic modes of vortex shedding behind the cylinders and observed the in-phase or anti-phase synchronization of wakes for specific gaps between cylinders. He observed a bistable biased flow pattern for critical spacing (1.1 / 2.2T D≤ ≤ ), i.e., narrow and wide wakes flip- flopped randomly behind each cylinder as time adva- nced. For gaps larger than 2.2D , he observed two distinct wakes that were generally in-phase but as the gap increased to over 5D , the flow around two cylinders behaved like the case of a single fixed cylinder.

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The flow pattern around different arrangements of circular cylinders at high Reynolds numbers was also studied experimentally by Zdravkovich[6]. In his work, he observed two different flow patterns around side-by-side cylinders ( 4= 6 10Re × ) due to the biased gap flow.

Guo et al.[7] considered the case of two rotating side-by-side cylinders. In their experimental study, they used the PIV method to analyze the vortex shedding behind these cylinders. They conducted their experiments for / = 1.1T D and 425 1130Re≤ ≤ .They recognized the importance of gap spacing in the wake pattern and shedding of vortices. They observed that after the utter suppression of vortex shedding at a critical rotational speed, the boundary layer separation disappears at higher speeds.

In the last decade, numerical approaches have been infrequently performed to numerically study the flow around two side-by-side cylinders. Slaout and Stansby[8] simulated the flow around two side-by-side cylinders at high Reynolds number (103-105) by using a mesh-free discrete vortex method which was based on potential flow and had reasonable results only for high Reynolds numbers.

In other works, Meneghini et al.[9] have employed a fractional step method to study the vortex shedding and wake interference between tandem and side-by-side cylinders at = 200Re at different spacings. In side-by-side arrangement, they observed the flip-flopping flow pattern for / 2T D ≤ and an anti- phase synchronized vortex shedding for / 3T D > .

Liu et al.[10] numerically investigated the wake pattern and flow behavior behind two side-by-side cylinders at low Reynolds numbers. They used an unstructured spectral element method for their work and found 9 different, i.e., four steady and five unsteady, wake patterns for various gap spacings ( / =T D1.1- 3.0 ) and Reynolds numbers (18 to 100). They reported that the wake patterns are strongly dependent on the distance between two cylinders and are not that sensitive to the Reynolds number.

Furthermore, Chen et al.[11] employed a finite volume method with unstructured mesh to study the formation and convection of vortices behind two side- by-side cylinders using a LES model. The simulations were done at = 750Re and / = 1.7, 3T D . They observed a deflection in gap flow for / = 1.7T D ,while at / = 3T D two symmetrical wakes were formed behind the cylinders.

According to the works mentioned above, it can be seen that due to the complex flow field around two- cylinder geometry, most of the cited numerical studies have focused on simulations at low Reynolds number and very few addressed the study of these arrangements in turbulent flow. Hence, in this paper a diffe-

rent numerical approach has been developed to solve and analyze laminar and turbulent flows around two side-by-side cylinders. The presented results in the current work provide a comprehensive analysis of flow characteristics and hydrodynamic force variations for two side-by-side circular cylinders in both flow regimes.

A Cartesian-staggered grid finite-volume based solver is developed for performing the simulations, and the identification of solid bodies in the flow field and implementation of no-slip condition are carried out by using great-source term and ghost-cell techni- ques, respectively. The grid is chosen finer where the cylinders are located and computational solution procedure is based on the SIMPLE algorithm. The solver has been previously developed and put into effect for solving flow over complex geometries by Ghadiri Dehkordi and Houri Jafari[12] Consequently, the governing equations and applied computational algorithm are summarized. Then the present computational code is validated through simulation of laminar/turbulent flows around a single rigid cylinder. After that computational results for two side-by-side cylinders at

=Re 100, 200 and 104 and different gap ratios are provided. Finally, the conclusions of current work are drawn.

1. Computational approach to the problem 1.1 Equations of the problem

The differential equations governing incompressible, viscous, unsteady and 2-D fluid flow comprise the continuity and momentum equations which are written as follows:

= 0i

i

ux

∂∂

, = 1,2i (1)

21+ = +i i ij

j i j j

u u uput x x x x

νρ

∂ ∂ ∂∂−∂ ∂ ∂ ∂ ∂

(2)

where iu the velocity components, p the pressure, ν the kinematic viscosity of fluid and ρ the fluid density. The density is assumed constant in calculations due to the incompressibility of fluid.

However, for turbulent regime one of the best- known and widely-used two-equation models, i.e., k ε− model, is employed to model the turbulence. Details about this model and its governing equations are comprehensively explained in Ghadiri-Dehkordi et al.[13].

It should be noted that the flow has been assumed incompressible with = 1 000ρ and 3= 1.002 10μ −×throughout the current study.

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1.2 Problem limitsA 40 30D D× rectangular domain is provided

for locating two equal-sized circular cylinders in side- by-side arrangement. The centers of both cylinders are located at 10D from the inlet boundary, 30D from the outlet boundary and 13 -14D D from upper and lower boundaries, where D denotes the cylinder diameter and has been chosen equal to 0.04. Flow enters from the left side of the field with the velocity of Uand coordinates of cylinders’ centers have been set at (10D ,10D ) .The computational domain for side-by- side arrangement is shown in Fig.1 where T is the transverse gap spacing between two cylinders.

Fig.1 Computational domain for two side-by-side cylinders

At the inlet of the flow field, the Dirichlet boundary conditions are applied ( =u U , = 0v , =k

20.03U and 2= / 0.005k lε , where U is the free- stream velocity, and l is the length of computational domain) and at the outlet, the Neumann-type conditions are employed ( / = 0u x∂ ∂ , / = 0v x∂ ∂ , /k∂

= 0x∂ and / = 0xε∂ ∂ ). However, regarding the enforcement of no-slip

condition on the curved boundaries of cylinders, in this study this condition is applied using the ghost-cell method. This method which has been described in detail by Ghadiri Dehkordi and Houri Jafari[12], offers an outstanding solution for the problem of fitting curved boundaries of the cylinders onto the grid points inside a Cartesian grid. Accordingly, to assign values to velocity components, the great-source term technique described by Ghadiri Dehkordi and Houri Jafari[12], is employed.

Fig.2 A close view of generated mesh for two side-by-side cylinders

2. Grid generation and discretization of the equations

In the current work, the grid generation in the flow field is done through a non-uniform Cartesian- staggered grid in order to implement the present computational code. In this grid, as is shown in Fig.2, the enclosed area of two cylinders is refined with respect to other areas of the flow field. A grid distribution of 210×210 cells is selected for the computations around the circumference of cylinders. To show the independency of results from the grid size, the verification of mesh independency was done for different grid sizes. The results are described in detail in Ghadiri-Dehkordi et al.[13]. According to this verification, for grids finer than 210×210 cells, no significant change was observed and there was just a 0.07% difference between this grid structure and 190×190 grid. So the solution is sufficiently grid independent and hereinafter other simulations of the paper are carried out by using the 210×210 cells grid.

Moreover, the pressure, kinetic energy and dis- sipation rate of turbulence ( p , k and ε ) are calculated for main grid points, while u and v velocity components are calculated for the points located on the staggered control volume surface.

The discretization of governing equations is carried out using the finite-volume method, where a fully implicit approach is applied for temporal discretization of equations and space discretization is performed using a hybrid scheme. The general forms of dis- cretized equations including momentum, pressure and velocity correction equations are presented in detail in Patankar[14] for reference, and hence, they are not mentioned again in here.

3. Solution procedure Finally, the solution procedure of current study is

based on the SIMPLE computational algorithm[14].Main steps of SIMPLE algorithm is summarized as follows: guess of pressure field, solution of momentum equations, calculation of the total residual ( b ),pressure and velocity correction, solution of other differential equations for φ (equations of k and ε )and iteration of all steps until the full convergence ( 0.0b ≈ ) is reached. This residual includes the resi- duals of momentum and k , ε equations and pressure correction residual from continuity equation. Hence, choosing a very small value, 10–6 in this paper, seems appropriate and adequate. As a matter of fact, this value represents the intrinsic error of discretization which is unavoidable. Considering the present approach for discretization of governing equations, lower values than this would have no significant effect on the accuracy of our solution.

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4. Validation of the solver Analysis of shedding process and flow behavior

behind a single cylinder would greatly help to under- stand the flow characteristics between two side-by- side cylinders. On the other hand, this benchmark has been investigated by many researchers both experimentally and numerically and is a proper case for validation of the solver.

Table 1 Comparison of mean drag coefficient,mDC , maxi-

mum lift coefficient,mLC and Strouhal number,

St , for a single cylinder at = 100Re

= 100Re

mDCmLC St

Present work 1.36 0.36 0.166

Exp. results Kang[15] 1.33 0.32 0.165

Liu et al.[10] 1.37 0.33 0.165

Sa and Chang[16] 1.23 0.35 0.155

Lee and Yang[17] 1.34 0.33 0.165

Num. results

Park et al.[18] 1.33 0.33 0.165

Fig.3 Time histories of lift and drag coefficients for the flow past a stationary cylinder at = 100Re

Fig.4 Flow characteristics around a single stationary cylinder at = 100Re and = 77tU/D

In this section, the current numerical approach and employed computational code are validated through simulation of flow around a fixed circular cylinder at = 100Re (laminar case) and the results are compared with other experimental and numerical works in Table 1. The comparison of results for the turbulent case is provided in Ghadiri-Dehkordi et al.[13]. The lift and drag coefficients and the Strouhal number are defined as follows:

2=

12

Ll

FCu Dρ

,2

=12

Dd

FCu Dρ

, = vf DSt

U mLC

where LF and FD denote lift and drag forces, respectively. St is the Strouhal number, D is the cylinder diameter, and vf is the frequency of vortex shedding, which can be calculated from the oscillation frequency of lift force.

Fig.5 Effective viscosity distribution for flow around a fixed circular cylinder at 4= 10Re and = 87tU/D

The simulation results of flow past a stationary cylinder are presented for mean drag coefficients, maximum lift coefficient and Strouhal numbers and are compared with the data in literature in Table 1. Non-dimensional time step has been set to 0.004. From Table 1, we can see that at = 100Re the resu-

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lts of the present work agree closely with other results. Time histories of lift and drag coefficients for this Reynolds number are shown in Fig.3, which is followed by pressure and vorticity contours for the mentioned Reynolds number in Fig.4. As it is evident, vortices are shed periodically from the cylinder and a vortex street is formed at these Reynolds numbers. The lift and drag coefficients also oscillate due to the asymmetric pressure distribution occurred by periodic vortex shedding.

Finally, effective viscosity distribution in the flow field around a fixed circular cylinder is calculated by all three different RANS models at 4= 10Reand the results are depicted in Fig.5. As it can be seen from this figure, the effective viscosity distribution simulated by the extended and RNG k ε− models show lower values with respect to the standard k ε−model, which implies that this case has a lower shedding. The separation region is also greater than that obtained with the standard model and is closer to experimental conditions. Due to this large separation region around a single cylinder and considering the features of mentioned models, this result is predictable. The prediction of larger separation region with modified models offers a greater pressure drop, which leads to an increased pressure lift coefficient. Hence, the drag coefficients calculated by modified models are greater than those of the standard model and are rela- tively close to experimental results presented by Ghadiri-Dehkordi et al.[13]. In this case, the most com- parative value of drag coefficient and the best simulation of effective viscosity distribution in accordance with the exact method of DNS are obtained by RNG model. Therefore, this model is employed for simula- ting the turbulent flow around two side-by-side cylinders.

5. Flow over two side-by-side cylinders The study of flow around a fixed cylinder pre-

sented in previous section provides a good insight into better understanding the flow characteristics around two-cylinder geometries. In this section, the results for the simulation of flow around two side-by-side circular cylinders are analyzed and comparison of these results with other data are presented to reveal the effect of different flow parameters on the flow pattern. It will be seen that the variation of the Reynolds number and transverse gap have significant effects on flow characteristics behind upper and lower cylinders. These changes lead to a notable deviation from the results obtained for a fixed cylinder in terms of hydrodynamic forces, wake pattern and vortex shedding from both cylinders.

Accordingly, first the laminar flows around two side-by-side cylinders are comprehensively studied and then it is followed by the results of turbulent

regime where the flow shows different behaviors due to high Reynolds number. 5.1 Laminar flow, Re = 100

In this section laminar flows around two side-by- side cylinders at = 100Re and gap ratios of / =T D1.5, 3 and 4 are simulated and temporal variation of hydrodynamic forces along with pressure and vorticity contours and streamlines are presented for each case.

Fig.6 Flow characteristics for two side-by-side cylinders at = 100Re = 1.5t/D and = 65tU/D

Fig.7 Flow characteristics for two side-by-side cylinders at = 100Re = 3t/D and = 65tU/D

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The pressure field around two side-by-side cylinders at / = 1.5T D is shown in Fig.6(a), where as a consequence of close proximity of cylinders a repulsive force acting on both cylinders is observed. This was also reported in the experimental work of Bearmann and Wadcock[19], and numerical work of Meneghini et al.[9]. As is shown in Fig.6(b), we can see that at / = 1.5T D as time advances, the wakes behind cylinders merge as a result of proximity of cylinders, while for larger gaps, / =T D 3, 4, according to Figs.7(b) and 8(b) these wakes are formed separately. However, at = 100Re and / = 1.5T D ,as one can see from Fig.6(c), there exists a bi-stable flow pattern behind cylinders, i.e., wide and narrow wakes are formed behind each cylinder which switch by the variation of time. This flow pattern is in good agreement with the results of Zdravkovich[9] who observed a biased flow pattern for the gap range of 1.1 / 2.2T D≤ ≤ . Also from Fig.6(c), it is observed that at this gap the flow is deflected toward one of the cylinders and changes its way toward the other one after some periods of vortex shedding. As was also predicted by Liu et al.[10] in their numerical work, the flip-flopping phenomenon is also observed in current work at / = 1.5T D (see Fig.6(b)). Due to flow deflection in gap flow, a narrow wake region is formed behind the cylinder toward which the flow is deflected, while a wide wake region is formed behind the other cylinder.

Fig.8 Flow characteristics for two side-by-side cylinders at = 100Re = 4T/D and = 65tU/D

Nevertheless, by increasing the transverse gap to / = 3T D , it can be seen from Figs.7(a)-7(c) that flow

characteristics show different behaviors due to the

change of gap. Figure 7(a) presents the pressure contours for this case. Figure 7(b) shows the asymmetric shedding of synchronized in-phase vortices. As time advances, a longer wake is formed in the vertical direction due to the merging of these vortices which stops at a distance downstream the cylinders. It can be seen from Figa.7(b), and 7(c) that unsteady flows around upper and lower cylinders separate from the surface of cylinders at counterpart points. As is shown in Fig.7(c), streamlines in the wakes of both cylinders follow an in-phase asymmetric pattern. Also, separation points of both cylinders occur near the right side of the base of each cylinder.

Finally, at / = 4T D (see Fig.8) a symmetric anti-phase flow pattern was found from the simulation of flow around side-by-side cylinders which showed good agreement with numerical results of Ding et al.[20] and Liang et al.[21]. This flow pattern can be seen in Fig.8(b). Two Karman vortex streets are clearly observed in this figure, as described by Zdravkovich[9].From Fig.8(c), it can be seen that separation points of flow are symmetric with respect to the centerline of gap spacing.

Fig.9 Time histories of lift and drag forces for = 100Re and = 1.5T/D

Time histories of lift and drag forces for different gap ratios at = 100Re are depicted in Figs.9 through 11. From these figures, we can clearly see that by decreasing the gap both cylinders experience higher values of lift forces. However, as the gap decreases the velocity of gap flow increases and therefore a more significant difference is predicted in the time- averaged variations of lift forces at lower gap ratios. The lift and drag coefficients vary irregularly with time at / = 1.5T D which implies that the flow is unsteady and non-periodic, as shown in Fig.9. As Zhou

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et al.[22] and Xu et al.[23] reported in their experimental work, gap flow deflects to the cylinder with higher frequency which leads to a higher drag coefficient for this cylinder. This was also observed in our paper and is shown in Fig.9. Also from this figure, we can clearly see that time varying drag coefficients at this gap change out of phase and as a result of random flip-flop of wakes behind cylinders in the direction of gap flow, both cylinders irregularly experience higher drag coefficients. Figure 9 shows that at this gap, the upper cylinder experiences a positive lift coefficient while this coefficient takes negative values for the lower cylinder. This is justified by noting that the transverse gap between cylinders inhibits the evolution of flow and consequently a high-pressure region is developed in the separation distance. On the other hand, due to higher shedding frequency for lower cylinder, it experiences higher drag values than does the upper cylinder. Accordingly, the flow deflects from lower cylinder to the upper one. As a consequence of this deflection, time-averaged variations of lift will be different for each of these cylinders.

Fig.10 Time histories of lift and drag forces for = 100Re and = 3T/D

In Fig.10, the time histories of hydrodynamic forces for / = 3T D are shown. From this figure we can observe that drag coefficient variations at this gap are similar but out of phase, the oscillation amplitudes of this force follow a similar trend but they have different values. Comparing Figs. 9 and 10, we can see that the difference in oscillation amplitudes for both cylinders at / = 3T D is lower than those at / = 1.5T D ,which is because of the diminishment of interference effects due to the increase of gap distance. From Fig.10, we can also observe that lift coefficients at this gap are in anti-phase with respect to each other but

they approach an in-phase trend after some periods of vortex shedding. Synchronized variations of lift and drag coefficients (see Fig.10) corroborate the periodic behavior of vortex shedding from both cylinders at this gap.


Fig.12 Flow characteristics for two side-by-side cylinders at = 200Re , = 2T/D and = 65tU/D

Finally, at / = 4T D the synchronized variation of hydrodynamic forces with time confirms the synchronized vortex shedding from both cylinders, as we can see in Fig.11. Similar to the observations at

/ = 3T D , we can observe from Fig.11 that the mean

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Table 2 Comparison of flow parameters for two side-by-side cylinders at = 100ReMean drag coefficient Mean lift coefficient Strouhal number Parameters

results Present Ding

et al.[20]Liu

et al.[10]Present Ding

et al.[20]Liu

et al.[10]Present Ding

et al.[20]Liu

et al.[10]

UC 1.54 1.53 1.47 –0.47 –0.46 –0.49 - - - / = 1.5T D

LC 1.52 1.51 1.45 0.47 0.47 0.47 - - -

UC 1.49 0.02± 1.56 0.03± 1.55 0.05± 0.13 0.05− ± 0.13 0.25− ± 0.19 0.02− ± 0.183 0.182 - / = 3T D

LC 1.49 0.02± 1.56 0.03± 1.55 0.05± 0.13 0.05± 0.13 0.25± 0.19 0.02± 0.183 0.182 -

UC 1.47 0.03± 1.51 0.03± - 0.064 0.01− ± 0.07 0.34− ± - 0.186 0.184 - / = 4T D

LC 1.47 0.01± 1.51 0.01± - 0.064 0.01± 0.07 0.34± - 0.186 0.184 -

drag coefficient is the same for both cylinders at this gap. The oscillation amplitudes of lift forces differ slightly from the previous case ( / = 3T D ), except that due to more increase in gap spacing, interference effect diminishes and therefore a lower difference in oscillation amplitudes of lift coefficients is observed.

A comparison of different flow parameters at = 100Re and different gaps with other numerical and

experimental results is provided in Table 2.

Fig.13 Flow characteristics for two side-by-side cylinders at = 200Re , = 3T/D and = 65tU/D

5.2 laminar flow, Re = 200 It is shown in current work that interference effe-

cts and flow pattern around two side-by-side cylinders are sensitive to the Reynolds number and gap spacing, with the latter having a stronger effect. The effect of increase in the Reynolds number is verified in more detail for = 200Re in this section. The pressure and vorticity contours and streamlines for different gaps at = 200Re are shown in

Figs.12-13. At / = 1.5T D , the flow shows a similar behavior to that of = 100Re for the same gap. Studying this figure reveals that a repulsive force acts on both cylinders at this gap, as described by Bear- mann and Wadcock[15] and Williamson[3] in their experimental work and Meneghini and Saltara[9] in their numerical study for flow around side-by-side cylinders. A high-pressure region developed in front of these cylinders. The pressure in this region drops as the flow passes the gap. The result of this pressure field is the occurrence of repulsive force. The wake pattern behind side-by-side cylinders at this gap is similar to the one behind a single cylinder case. Liu et al.[10] reported a similar pattern in their numerical study for small spacing ( / 1.5T D ≤ ). This behavior is well justified by introducing proximity effect occuring at small separations in this arrangement, which is a case for 1 / 1.1-1.2T D< < that Zdravkovich reported in his experimental study. Two cylinders lying close to each other will affect their motion, due to the effects of interference on the hydrodynamic pressure field around them. Due to this effect the wakes behind both cylinders merge into a single wake and reproduce the single-body behavior.

It was mentioned before that for this gap, the biased flow pattern is bi-stable, i.e., narrow and wide wakes behind both cylinders as well as the direction of gap flow switch by time from one cylinder to the other. Wake patterns behind these cylinders do not show a synchronized behavior at this spacing.

As the gap spacing increases to / = 2T D , the characteristics of flow show a slight difference due to the decrease of interference effect. From the pressure and vorticity contours given in Figs.12(a) and 12(b) respectively, we can observe that despite the diminishment of repulsive force between cylinders, the wake still holds the pattern of a single cylinder. The wake does not follow a regular pattern and flip-flopping phenomenon still exists. Regarding the suction pressure around lower cylinder, it can be seen from Fig.12(b) that vortex shedding from this cylinder is stronger than the one from the upper cylinder. The

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streamlines for this case, as shown in Fig.12(c), follow a similar pattern as the ones at / = 1.5T D where for the present case the gap flow deflects toward the lower cylinder. Studying the presented contours reveals a close similarity between current flow pattern with the one at / = 1.5T D , except that unlike previous case the vortices are merged nearly outside the separation distance at / = 2T D .



However, at / = 3T D the flow pattern is different from the patterns observed for the two previous gaps. Due to the increase of the Reynolds number, at this gap pressure is distributed differently in comparison to the same gap at = 100Re , as shown in

Fig.13(a). The flip-flopping pattern no longer exists and as shown in Fig.13(b) a synchronized anti-phase flow pattern is observed behind both cylinders, vortices are not merged and begin to shed separately as a single vortex street. From Fig.13(c), the symmetry of flow field with respect to the gap spacing between cylinders can be observed. Comparing the flow behaviors for the counterpart gaps at this Reynolds number, we can clearly see the effect of the Reynolds number on flow characteristics. On the other hand, the presented contours and streamlines confirm that the unsteady flow around two cylinders shows more sensi- tivity to the change of separation distance. Conse- quently, an asymmetric wake pattern is observed behind each cylinder at this gap, as shown in Fig.13(b).

Finally, at / = 4T D the flow shows similar behavior to the previous case. However, the repulsive force between the cylinders is completely diminished and a fully periodic vortex shedding is observed from both cylinders. The flow pattern becomes more symmetric than before because interference effects decrease significantly as the gap increases. The asymmetric wakes behind upper and lower cylinders are in anti-phase.

Fig.16 Flow characteristics for two side-by-side cylinders at 4= 10Re , = 2T/D and = 65tU/D

The time histories of lift and drag forces at = 200Re and / =T D 2 , 4 are depicted in Figs.14

and 15. At / = 1.5T D , the hydrodynamic forces vary irregularly with time like the similar case at = 100Re .No repeatability is predicted in the variations of these coefficients and therefore they follow a certain pattern. The oscillation amplitudes and mean values of drag

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Table 3 Comparison of flow parameters for two side-by-side cylinders at = 200ReMean drag coefficient Mean lift Coefficient Strouhal number Parameters

results Present Ding

et al.[20]Meneghini[9] Present Ding

et al.[20]Meneghini[9] Present Ding

et al.[20]Meneghini[9]

UC 1.31 1.54 1.32 –0.36 –0.41 –0.40 - - - / = 1.5T D

LC 1.30 1.52 1.32 0.36 0.43 0.40 - - -

UC 1.42 - 1.42 –0.21 - -0.22 - - - / = 2T D

LC 1.41 1.56 0.03± 1.42 0.21 - 0.22 - - -

UC 1.38 1.548 1.41 –0.12 –0.104 –0.10 0.213 0.215 - / = 3T DLC 1.38 1.548 1.41 0.12 0.104 0.10 0.213 0.215 -

Note: UC is the refers to the upper cylinder, LC is the refers to the lower cylinder.

coefficients are the same for both cylinders and the pattern of these oscillations corroborate the flip-flopping phenomenon. As the flow is deflected toward either of two cylinders, drag forces increases. Irregular variation of lift forces reveals the proximity effect between cylinders due to the small spacing.

At / = 2T D , temporal variations of lift and drag forces are still irregular, where drag force variation represents the flip-flopping pattern and lift force variation demonstrates the net effect of the repulsive force between cylinders, as shown in Fig.14. Ampli- tude and intensity of lift coefficient and the mean value of drag coefficient increase with respect to the previous case, but the mean value of lift coefficient decreases as the gap increases. It can also be seen that for this gap, / = 1.5T D , a positive mean lift value is experienced by upper cylinder while the lower one takes negative values of lift coefficient. The reason is that while the pressure inside the gap flow is still higher than the pressures at the opposite sides of each cylinder, the pressure drops in the gap flow from the high-pressure region in front of the cylinders to the low-pressure region.

However, by increasing the gap to / = 3T D the oscillation pattern of hydrodynamic forces shows a significant change with respect to the similar case at

= 100Re . As is shown in Fig.15, due to the occurrence of two anti-phase wake patterns behind both cylinders (see Fig.13), the synchronization of lift forces from upper and lower cylinders is observed which confirms the synchronized behavior of vortex shedding for this spacing. On the other hand, the anti- phase drag forces are the same for both cylinders and their mean values increase again.

Finally, at / = 4T D , the flow shows similar behavior to the one at / = 3T D except that the mean value of lift coefficient in both cylinders decreases while the mean drag coefficient takes higher values. The lift forces of cylinders are in anti-phase and their mean values are lower than the ones obtained for previous gaps.

A comparison of different flow parameters at

= 200Re and different gaps with other numerical and experimental results is presented in Table 3. Generally, it is worth noting that at a given gap, the Strouhal number increases with the increase of the Reynolds number. Due to the existence of unsynchro- nized wakes for small gaps, i.e., / =T D 1.5, 2, which results from the flopping regime at this gaps, the FFT analysis on lift oscillations shows a spectrum with a broad band without a specific sharp peak. Hence, the Strouhal numbers are not provided for these gaps. 5.3 Turbulent flow, 4Re = 10

The flow behavior along with characteristics of flow around side-by-side cylinders in laminar flow were discussed and analyzed in the previous sections. Now, we will study the effect of increasing the Reynolds number to subcritical values on the flow over two cylinders in transverse arrangement. As was mentioned in Section 2, each turbulence model sati- sfies specific conditions. Comparison of obtained results for a single circular cylinder with different k ε− turbulence models reveals that among the three mentioned models, the RNG k ε− model is more appreciable for the present condition of side-by-side cylinders in which separation of flow and shedding of vortices occurs.

Generally, in comparison to the experimental works of Zdravkovich[10], Akili et al.[24] and Sumner et al.[5], the present results provide similar flow patterns and mean values of hydrodynamic forces. Although most of these experimental works for turbulent regime have considered higher values of the Reynolds number than the present work, but our study show consistent results with theirs in terms of flow patterns and characteristics of lift and drag forces.

It is worth mentioning that, 2-D simulations of turbulent flow around bluff bodies at high Reynolds numbers do not seem accurate. As Williamson[25]

suggested in his work, the flow past bluff bodies show 3-D attributes for the Reynolds numbers greater than 180. With respect to the fact that flow around these bodies becomes turbulent around = 400Re , it is con- cluded that for high Reynolds numbers, 3-D effects

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should be considered in the calculations and hence the grid should be significantly fined which leads to costly calculations. Accordingly, implementation of 2-D methods has attracted much attention. It was observed that using this model with the present code gives reasonable results for mean values and fluctua- tion amplitudes of the forces.

The simulations were conducted at 4= 10Re for three different gap ratios of / =T D 2, 3 and 4 by using the RNG k ε− model. The non-dimensional time step is set to 0.006. Present simulations for turbulent flow have not been previously presented and analyzed by other researchers.

The vorticity, pressure and stream-function fields for / =T D 2, 4 are shown in Figs.16 and 17, respectively. The pressure field for / = 2T D , shown in Fig.16(a), suggests that both cylinders experience different base pressures. The predicted biased flow pattern is asymmetric and wake structures behind both cylinders are nearly similar, as shown in Fig.16(c). In accordance to the visualizations of Akili et al.[20] and Sumner et al.[5] for lower gaps than / = 2T D , we also can see that the effect of gap flow is less dominated due to the decrease of proximity effect. In-phase synchronized vortex shedding is obtained for this spacing (see Fig.16(b)) and wide and narrow wakes behind upper and lower cylinders intermittently change over with time and merge over time. According to Fig.16(b), the biased flow pattern pertains to the fact that the upper cylinder has a shorter near-wake region than that of the lower cylinder, as described by Sumner et al.[5] in their work for = 1900Re and same spacing.

At / = 3T D , a synchronized anti-phase vortex shedding occurs. On the other hand, as the gap increases the biased flow vanishes and less wake interaction between cylinders is observed, which implies that both cylinders behave independently. As was reported by Williamson[3], in this paper we also predicted that although in-phase behavior is also obse- rvable but the anti-phase behavior for vortex shedding in both laminar and turbulent regimes is predominant. Like the similar case at = 200Re , the formation of two Karman vortex streets is completely obvious for this case, but the turbulent pressure field around two side-by-side cylinders shows stronger pressure flu- ctuation, which is due to the increase in the Reynolds number. The synchronized symmetric flow pattern confirms the anti-phase behavior of cylinders at this gap ratio.

Finally, at / = 4T D more symmetric flow pattern is developed behind both cylinders due to the smaller interference effect, as shown in Fig.17(c). On the other hand, the synchronization of shed vortices still holds for this gap and unlike previous case, and we can observe from Fig17(b) that these vortices do

not merge at a distance from downstream cylinder. The pressure field around side-by-side circular cylinders at this gap is shown in Fig.17(a). Sumner et al.[5]

reported in their paper that for gaps greater than / = 4.5T D , no synchronization is observed in the

flow behavior.

Fig.17 Flow characteristics for two side-by-side cylinders at 4= 10Re , = 4T/D and = 65tU/D

Fig.18 Time histories of lift and drag forces for 4= 10Re and = 2T/D

The time histories of lift and drag forces for / =T D 2, 4 and 4= 10Re are shown in Figs.18 and

19, respectively. From these figures, we can observe that increasing the Reynolds number has a significant

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effect on the attributes of these forces. Accordingly, from the time histories of lift and drag coefficients in Fig.18 we can perceive that a vivid discrepancy exists for mean drag and lift of both cylinders at / = 2T Dand these mean values are nearly the same for both cylinders. Two reasons can be provided: Firstly, a weaker biased flow exists at this gap ratio, and secondly, during short irregular time intervals this biased flow switches into the opposite direction. The irregular variation of drag coefficients corroborates the flip-flopping phenomenon at this spacing.

Furthermore, due to the symmetric flow pattern at / = 3T D , the upper and lower cylinders show similar trends and symmetric behaviors in terms of drag and lift variations. Corresponding to the anti- phase vortex shedding at this gap ratio, the hydrodynamic forces of side-by-side cylinders follow an anti-phase trend at our Reynolds number ( 4= 10Re )while at higher Reynolds numbers there is a probabi- lity of in-phase behavior between cylinders. There is a difference in the oscillation amplitudes of lifts and drags for the upper and lower cylinders due to the interference effect but their mean value decays toward same small values.

Fig.19 Time histories of lift and drag forces for 4= 10Re and = 4T/D

Finally, From Fig.19 we can clearly observe that as the gap increases to / = 4T D the interaction between cylinders decreases noticeably, causing the lift forces approach to zero mean values for higher gap ratios. At this spacing, proximity effect diminishes due to the larger spacing between cylinders, both cylinders tend to behave as a single bluff body where their mean value of force coefficients confirms this tende- ncy. As is shown in Fig.19, we can see that unlike previous spacing ratios, at / = 4T D lift and drag

coefficients vary regularly with time as the wake interaction between upper and lower cylinders decreases significantly. The values of different flow parameters for turbulent regime are presented in Table 4. As one can clearly see, by increasing the gap both mean drag and lift values decrease while the Strouhal number increases.

Table 4 Different flow parameters for two side-by-side cylinders at 4= 10Re in present work

Parameters DC LC St

UC 1.37 0.22 - / = 2T D

LC 1.37 –0.22 -

UC 1.24 0.05 0.22 / = 3T D

LC 1.24 –0.05 0.22

UC 1.12 0 0.25 / = 4T D

LC 1.12 0 0.25

6. Concluding remarksIn this paper, a control-volume based solver has

been implemented to simulate the flow over two transverse cylinders in laminar and turbulent flows. Diffe- rent Reynolds numbers and gap ratios have been used in order to investigate the effect of various parameters on flow characteristics and hydrodynamic forces acting on the bodies. Resultantly, the most important conclusions of the present computational simulation of flow over side-by-side cylinders in laminar flow can be summarized as follows:

(1) At = 100Re , a biased flow pattern is observed behind both cylinders at / = 1.5T D , which is bi- stable. Narrow and wide wake regions develop behind each of the cylinders, which change over by time. Also, in-phase vortex shedding is observed for this gap and flip-flopping phenomenon is clearly reprodu- ced. Both lift and drag coefficients of cylinders variy irregularly with time due to the proximity effect. The gap flow is deflected toward the cylinder with higher frequency, which causes this cylinder to experience higher drag.

(2) For higher spacing ratios, / =T D 3, 4, at = 100Re the flip-flopping pattern diminishes as the

gap increases. A synchronized anti-phase vortex shedding behind upper and lower cylinders is predicted for / = 4T D , while at / = 3T D the shedding process and streamlines show an in-phase asymmetric pattern. Since lower cylinder has a higher drag, flow is deflected from this cylinder toward the upper one. The oscillation amplitudes of these cylinders show a slight difference for both / =T D 3, 4, where this difference is smaller for the latter due to the diminishment of interference effect. The mean values of these forces

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are the same at these gap ratios. (3) At = 200Re , the flow shows similar beha-

vior to that at = 100Re for / = 1.5T D . A repulsive force is developed between cylinders and the wake pattern behind cylinders is similar to that behind a single bluff body. Also, for this gap ratio due to the pressure difference between gap flow and opposite sides of each cylinder, the upper and lower cylinder experience positive and negative lifts, respectively.

(4) At = 200Re and / = 2T D , although the repulsive force between cylinders vanishes but the wake holds the similar pattern of a single cylinder. The wake follows an irregular pattern and flip-flopping phenomenon is observed. Increasing the gap to

/ = 3T D causes the flip-flopping phenomenon to diminish and a synchronized anti-phase wake pattern is observed. At / = 2T D , the force coefficients show irregular variation with time like the previous case and the mean value of lift decreases while that of drag increases. As the gap increased to / =T D 3, 4, the mean values of lift and drag decrease and both coefficients follow a certain pattern, implying the synchronized anti-phase pattern at these gaps.

(5) At = 200Re , a fully periodic vortex shedding is observed for / = 4T D and symmetric flow patterns are seen for the higher gaps.

In turbulent regime ( 4= 10Re ), the significant conclusions of the present simulation are:

(1) For / = 2T D , a biased asymmetric flow pattern is predicted and wide and narrow wake regions behind upper and lower cylinders intermittently change over with time. At / = 3T D , less wake interaction between cylinders is observed and a synchronized anti-phase vortex shedding is seen for this gap. At / = 4T D this interactions decreases and a more symmetric flow structure is observed in the streamlines.

(2) The mean values of drag and lift coefficients are discrepant and follow an irregular pattern for

/ = 2T D while at higher gap ratios ( / =T D 3, 4) they follow a certain pattern and their variations confirm the anti-phase vortex shedding where due to the diminishment of proximity effect at / = 4T D , the behavior of force coefficients are more symmetric at this gap ratio.

Finally, it is worthy to mention that the implemented solver is sufficiently applicable for simulation of flow over complex geometries such as multiple circular cylinders. Also, the presented results can be a good basis for prediction of structural responses in the flexible two side-by-side circular cylinders case.

Also, one can see that mean drag values for the current approach is lower compared to other methods. It should be noted that current approach uses a finite volume method to solve the flow field and hence,

leading to more accurate results which may be closer to real values.

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numerical simulation of flow over two side-by-side circular cylinders

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