Numerical investigation of flow around an inline square cylinder array with different spacing ratios

Computers & Fluids 55 (2012) 118–131

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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d

Numerical investigation of flow around an inline square cylinder array with different spacing ratios Yan Bao a, Qier Wu a, Dai Zhou a,b,⇑ a b

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, No. 800, Dongchuan Road, Shanghai 200240, China State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, No. 800, Dongchuan Road, Shanghai 200240, China

a r t i c l e

i n f o

Article history: Received 30 April 2011 Received in revised form 2 July 2011 Accepted 19 November 2011 Available online 26 November 2011 Keywords: Inline square cylinder array Flow pattern Wake interference Aerodynamic characteristics Spacing ratio Drag and lift coefficients Strouhal number

a b s t r a c t Flow around an inline cylinder array consisting of six square cylinders at a Reynolds number of 100 is investigated numerically by using a second-order characteristic-based split finite element algorithm in this paper. The numerical method and the code for the solution of incompressible Navier–Stokes equations are validated for the flow past a single and two tandem square cylinders, and the numerical results show a good agreement with the available literatures. The study then focuses on the effect of spacing ratio (ratio of center-to-center distance s to cylinder width d, widely ranging in s/d = 1.5–15.0) on flow characteristics by identifying flow patterns and extracting pressure distributions, force statistics as well as wake oscillation frequencies. Numerical results showed six different flow patterns, which appeared successively with the increase of gap spacing, namely, steady wake, non-fully developed vortex street in single row and double-row, fully developed vortex street in double-row, fully developed vortex street in partially recovered single-row and fully developed multiple vortex streets. A shielding effect of the first cylinder and reducing Bernoulli effect on the rear cylinder rows work in the pressure distribution even at very large gap spacing. In the vortex shedding regime, beyond the critical spacing of wake mode transition, force statistics show a periodic variation characteristic for the last four cylinders; moreover, multiple frequency components involve in the vortex shedding oscillation behind these cylinders and the dominant frequency jumps down with the increase of the spacing. Finally, the flow fields around the critical spacing range are comprehensively analyzed to reveal the crucial mechanism behind the observed aerodynamic characteristics. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The unsteady flow around an array of cylinders is of practical engineering importance, such as flow around buildings, wind turbine farms and chimney stacks. The flow over multiple cylinders has much richer fluid dynamic features [1–5]. The aerodynamic forces and vortex shedding frequencies as well as wake patterns are very different from those of an isolated cylinder, even at the same Reynolds number [6,7]. Sumner et al. [8] investigated experimentally two equal circular cylinders in staggered arrangement and identified nine flow patterns depending on the incidence and distance between the cylinders. Bradshaw [9] carried out some experiments on a row of nine circular cylinders in order to find out the stability limit for the merging of vortices at Re = 1500. Cheng and Moretti [10] investigated experimentally the wake of a row of nine tubes at Re = 200 with a gap ratio (the ratio of the separation between the cylinders and the width) of 3. Le Gal et al. [11] presented ⇑ Corresponding author at: School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, No. 800, Dongchuan Road, Shanghai 200240, China. E-mail address: [email protected] (D. Zhou). 0045-7930/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2011.11.011

flow-visualization and velocity measurements of the wake behind a row of cylinders placed side-by-side at Re = 80 with the spacing of 0.5d and 2d (d is the cylinder diameter). Igarashi and Suzuki [12] performed experiments on flow around three cylinders arranged in-line while Lam and Cheung [13] investigated the flow characteristics and interference effect of three cylinders in different equilateral arrangements. Lam et al. [14] measured the force coefficients and Strouhal numbers on four cylinders in a square configuration at subcritical Reynolds numbers in a wind tunnel. Ziada and Oengören [15] carried out experiments to study the vortex shedding characteristics in an inline tube bundle with different spacing. Recently, Liang et al. [3] investigated numerically the vortex shedding characteristics of laminar flow past an inline tube array. The effect of spacing ratio was mainly studied, but with a very limited spacing range. Their research showed that with the increasing of the spacing ratio, the flow becomes more asymmetric and vortex shedding is induced from the last cylinder then propagates toward upstream. Flow around square cylinders can be expected to has their own characteristics and be different from that of circular cylinders, since the square cylinders have a tendency to fix the separation points at one of the section edges. Investigations of flow past two square cylinders have been reported in the literatures in the past [16–20].

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Among others, Mizushima and Takemoto [21] investigated numerically the stability of flow over a row of square bars with the assumption that the flow was two-dimensional and incompressible. They found that vortex shedding is nearly independent when the spacing ratio is large while the confluence of several jets appears for small spacing ratio. For some combination of Reynolds number and spacing ratio, both flopping and bi-stable flip-flop behaviors were observed in the wake. Mizushima and Akinaga [22] investigated both numerically and experimentally the interaction of wakes of a row of square bars perpendicular to an incoming uniform flow. They observed in-phase vortex shedding for gap ratio 1.0 and anti-phase shedding for 3.0. Recently, Kumar et al. [1] investigated numerically the flow around a row of nine square cylinders placed normal to the oncoming flow for gap ratio from 0.3 to 12.0 at Re = 80. The flow regimes such as synchronized, quasi-periodic and chaotic have been identified for gap ratio smaller than 6.0, which are resulted from the interaction between primary and secondary vortex shedding frequencies. Chatterjee et al. [2] performed a numerical study for the flow around five square cylinders placed side-by-side and normal to the oncoming flow at Re = 150. They identified the flow patterns as flip-flopping pattern, in-phase and anti-phase synchronized pattern and non-synchronized pattern. The literature survey shows that not much attention has been paid to an inline cylinder array consisting of multiple (more than two) square cylinders, in particular, the flow characteristic has not been studied yet for a wide range of gap spacing. In Liang et al. [3], the spacing ratio is limited in the range from 2.1 to 4.0. This limitation is however not there in this work. In our investigation, the effect of spacing ratio in the range 1.5 6 s/d 6 15.0 is documented in terms of aerodynamic characteristics and the ensuing flow patterns, and further examines how the flow mechanism actually works for those phenomena. The motivation of current study is of a fundamental nature, yet it also relevant to some practical engineering applications such as large building blocks, heat exchangers tube bundles, and electronic devices. For example, in the design of high-rise building structures, the spacing distance of adjacent structures is a very important parameter due to the fact that shedding frequency and the aerodynamic forces may differ significantly for different spacing regimes. To get reliable knowledge of design parameters, such as drag and lift coefficients, vortex shedding natural frequency, and wake size, understanding of basic fluid mechanics that occur in multiple bluff bodies is fundamental prerequisite in the design. In the present study, for the representation of physical reality of exceedingly complex multi-cylinder configurations, more than two cylinders are needed; therefore, we follow Liang et al. [3] to use six cylinders in tandem arrangement. This work is organized as follow: Section 2 presents the governing equations for incompressible viscous flow and provides details on the employed numerical approach for the solutions. For the validation of the computer code and to provide a reference for further study, the results of laminar flow around a single cylinder and two tandem square cylinders with different spacing ratios are presented in Section 3. In Section 4, results for an in-line square cylinder array consisting of six cylinders with different spacing ratios are presented and discussed. Main findings from the present work are summarized in Section 5.

ð2Þ

A developed second-order CBS finite element algorithm [23] is employed to solve the above equations. In this algorithm, the form of characteristic-based semi-discretized scheme [24] is obtained by treating the time discretization along the characteristics explicitly, while the spatial discretization for the CBS algorithm is performed with standard Galerkin procedure [25] after the temporal discretization. In the framework of incremental projection method, the second-order CBS scheme requires an additional effort to be made to handle the pressure instability. In Bao et al. [23], two versions of pressure stabilization techniques were integrated into the CBS scheme. One uses T4/C3 MINI element to approximate both the velocity field and pressure field, the other is pressure gradient projection based stabilization method allowing to use equal order interpolation functions for both velocity and pressure. The former is employed in this paper, where the T4/C3 MINI element is obtained by adding a cubic bubble to a piecewise linear approximation of velocity while pressure remains piecewise linear in each triangular element. As compared with the traditional linear triangular element, the T4/C3 is more capable to approach the realistic physics of flow by taking into consideration of the central physical behavior of element. A Bi-conjugate gradient (BiCG) method was used for the solution of pressure Poisson equation, while velocity fields were explicitly obtained from previous time step. An assumption of laminar flow is valid at this considered low Reynolds number; therefore, the present numerical simulation is performed in two dimensional space, starting from zero velocity and pressure fields. The simulation runs until the flow develops fully with adequate length of data for the statistical analysis. The parallel programming of the simulation code based on the Message-Passing Interface (MPI) standard is unavailable at this time, but it is expected that a substantial improvement of computational efficiency will be realized by using of parallel BiCG iterative method based on the element by element (EBE) technique for the solution of pressure Poisson equation. The simulations were carried out on a PC with Intel Quad-core processor i7-630 (2.80 GHz) and 6 GB of RAM. The simulation takes around 71 h for the case of s/d = 5.5 on the grid system containing 133,498 cells, where the total nondimensionalized time length T = 500 with the time step size of 0.005. Some global flow parameters involved in the discussions, including drag coefficient, CD, lift coefficient, CL, Strouhal number, St, and pressure coefficient, Cp, are defined as following,

CD ¼

2F D U 21 d

q

;

CL ¼

2F L

q

U 21 d

;

St ¼

fs d ; U1

Cp ¼

2ðp  p1 Þ

qU 21

ð3Þ

where FD and FL are the force components in the stream-wise and transverse directions, respectively; fs is vortex shedding frequency determined from the power spectrum analysis of the fluctuating lift force; q is the fluid density, U1 and p1 are characteristic velocity and pressure of the problem, respectively; and d is the characteristic length scale and taken here as the cylinder width. Here, d and U1 takes the unit value respectively, so the St in Eq. (3) is equal to the vortex shedding frequency. 3. Validation study

2. Governing equations and numerical details The non-dimensional governing equations for incompressible viscous fluid flow moving in a domain X and in a time interval [0, T] are written in tensorial Cartesian form as:

@ui @ui @p 1 @ 2 ui ¼ þ þ uj @xi Re @xj @xj @t @xj

@ui ¼0 @xi

ð1Þ

3.1. Flow around a single square cylinder Here, the unsteady flow over a single square cylinder at Reynolds number of 100 is simulated to serve as a reference for further investigation of multiple cylinders. A computational domain of X = [20d, 35d]  [25d, 25d] is used for simulation and the inlet boundary is located 20d upstream from the center of the cylinder

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and the outlet boundary 35d downstream, the upper and lower boundaries are located 25d away from the horizontal centerline of the computation domain. At the inlet boundary, the streamwise and transverse velocities are set to 1 and 0: u = 1, v = 0, respectively, while at the exit an outlet boundary is prescribed as ou/ ox = 0, ov/ox = 0, p = 0. This boundary condition is valid only when the flow is fully developed, but it is permissible provided that the outlet is located far downstream from the region of interest [2,33]. A slip boundary condition is imposed on the lateral boundaries: ou/oy = 0 and v = 0. No-slip conditions are applied for cylinder surface: u = 0, v = 0. Computational results of some global parameters, including mean drag coefficient, root mean square (r.m.s.) value of lift coefficient and Strouhal number, are compared and show excellent agreement with both the experimental and numerical data in existing literatures, see Table 1.

∂u/∂y=0, v=0

∂u/∂x=0 ∂v/∂x=0 p=0

x

∂u/∂y=0, v=0 Fig. 1. Schematic diagram of the computational domain and boundary conditions for flow past two tandem square cylinders.

3.2. Flow around two tandem square cylinders For the second validation case of flow past two tandem square cylinders (Re = 100), the detailed information of the computational domain and boundary conditions are scheduled in Fig. 1. The inlet is located 20d upstream of the front cylinder (denoted with cylinder 1), while the outlet is placed 35d downstream of rear cylinder (cylinder 2). The upper and lower boundaries are placed 25d away from the horizontal centerline of the cylinders. The boundary conditions are the same with those used for the above single square case. Fig. 2 shows the computational mesh with spacing ratio s/d = 5.0. A rectangular mesh refinement box is used around two cylinders for more efficient calculation. Another refinement box is also used in the wake to prevent the size of mesh from growing quickly to the outlet. Spacing ratios s/d from 1.25 to 11.0 are examined. Firstly, a grid dependence study is carried out for three different meshes (coarse, medium and fine) with spacing ratio s/d = 5.0. The grid nodes distributed uniformly along the surface of the cylinder are 120, 160, and 200, respectively. The results of the mean drag coefficient, r.m.s. value of lift coefficient and the Strouhal number obtained for three cases are summarized in Table 2, where the discrepancies in percentage between the results from coarse and fine grids, as well as those between medium and fine grids are also shown. It is noted that the maximum difference between medium and fine grids is 0.35% in the value of r.m.s. lift coefficient for cylinder 1. In fact, the results from Breuer et al. [30] indicated that a higher resolution is required only at larger Reynolds numbers. However, the computation for the fine grid case takes much more time than that for medium grid case. Due to the noticeable increase of computational expense for fine grid, the results to be presented later are all based on the medium grid system. The variation of force coefficients with spacing ratio for two cylinders is illustrated in Fig. 3, where the results of a single cylinder and Lankadasu and Vengadesan [20] are also presented. The mean drag coefficients of both two cylinders are smaller than that of a single cylinder as observed in Fig. 3a, clearly evidencing the effect of spacing ratio on the drag force. For cylinder 1, it decreases with the increasing of spacing ratio, obtains its minimum value at s/

Table 1 Comparison of global parameters with the values from literatures at Re = 100. Source

CD

C 0L

St

Okajima exp. [26] Norberg exp. [27] Sharma and Eswaran [28] Sahu et al. [29] Present

– – 1.494 1.488 1.493

– – 0.192 0.188 0.180

0.141 0.143 0.149 0.149 0.145

Fig. 2. The computational mesh for flow past two tandem square cylinders with s/d = 5.0.

Table 2 Comparison of the global parameters for the grid independence study. Grid

Coarse Medium Fine

Cylinder 1

Cylinder 2

CD

C 0L

St

CD

C 0L

St

1.436 (0.70%) 1.428 (0.14%) 1.426

0.291 (0.59%) 0.290 (0.35%) 0.289

0.130 (0) 0.130 (0) 0.130

1.104 (0.45%) 1.098 (0.09%) 1.099

1.190 (1.73%) 1.209 (0.17%) 1.211

0.130 (0) 0.130 (0) 0.130

d = 3.75 and shows a sudden jump at s/d = 4.75, then continues to increase slowly and shows a tendency to approach the value of a single cylinder. For cylinder 2, the presence of negative drag coefficient can be observed clearly for s/d 6 4.25, which indicates that cylinder 2 is pushed upstream by fluid. Such feature has been reported by Zdravkovich [6] among others, where it also jumps at s/d = 4.75 but does not approach that of a single cylinder even for large spacing ratio. The variation of r.m.s. drag coefficients with spacing ratio for both two cylinders shows that, for small spacing ratio, the r.m.s. drag coefficients of both two cylinders are very small due to the stagnant

Y. Bao et al. / Computers & Fluids 55 (2012) 118–131

121

Fig. 3. Variation of force coefficients with the spacing ratio for flow past two square cylinders in tandem arrangement: (a) mean drag coefficient; (b) r.m.s. value of drag coefficient; (c) r.m.s. value of lift coefficient.

vortex pair between two cylinders (Fig. 3b). Both of them increase sharply between s/d = 4.5 and 4.75 and reach their maximum values at s/d = 4.75, after then drop to approach the level of a single cylinder. But the r.m.s. drag coefficient of cylinder 2 begins to increase slowly after s/d = 7.0. Similar to the r.m.s. drag coefficients, the r.m.s. lift coefficients for both cylinders are very small for s/d 6 3.75, see Fig. 3c. A sudden jump appears between s/d = 4.5 and 4.75. When s/d > 4.75, the r.m.s. lift coefficient of cylinder 1 decreases to reach the value of a single cylinder whereas that of cylinder 2 continuously increases to obtain its maximum value at s/d = 6.0 then begins to decrease. However, the r.m.s. lift coefficient of cylinder 2 is always higher than that of cylinder 1, due to the wake interference of cylinder 1 on the downstream cylinder 2. The Strouhal number, St, is determined from the power spectrum analysis of the fluctuating lift forces with a power spectral density (PSD) plot. The variation of Strouhal number against the spacing ratio is represented in Fig. 4. In general, the Strouhal number decreases with the increasing of spacing ratio in the small spacing regime (s/d 6 4.5), then increases to approach the value of a single square (4.75 6 s/d 6 11). A sudden jump is observed between s/d = 4.5 and 4.75. It can be seen that the Strouhal number remains constant for some spacing ratio intervals, such as 5.0 6 s/d 6 8.0. The experimental data of Huhe-Aode et al. [31] for two cylinders in tandem arrangement also showed that the Strouhal number stays constant for 5.0 6 s/d 6 10.0. Similar to the results of Sharman et al. [32] and Lankadasu and Vengadesan [20], the Strouhal number is found to be the same for both cylinders with all spacing ratios considered in present study.

4. Flow around an inline square cylinder array After the validation study, attention is now focused on the problem of flow around inline cylinder array that consists of six square

Fig. 4. Variation of the Strouhal number with spacing ratio.

cylinders. The cylinders of the same width, d, with the same spacing distance, s, between adjacent ones, are subjected to unconfined uniform flow. The considered spacing ratio is ranged widely in 1.5 6 s/d 6 15.0. The Reynolds number based on the undisturbed upstream velocity and the cylinder width is fixed to Re = 100. Fig. 5 shows the detailed information of the computational domain and boundary conditions for this multiple cylinder configuration. The inlet is located 20d upstream of the first cylinder (denoted with cylinder 1), while the outlet is placed 30d downstream of the last cylinder (denoted with cylinder 6). And the slip upper and lower boundaries are placed 20d away from the horizontal centerline of the cylinders. The boundary conditions described for the flow over a single square case are applied in this simulation. Fig. 6 shows the computational mesh with the spacing ratio s/ d = 5.0. Rectangular mesh refinement boxes are used around each cylinder and in the wake region behind the cylinder bank, the grid density is increased properly for the adequate resolution of the wake evolution.

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∂u/∂y=0, v=0

∂u/∂x=0 ∂v/∂x=0 p=0

∂u/∂y=0, v=0 Fig. 5. Schematic diagram of the computational domain and boundary conditions for flow past an inline square cylinder array.

In the past, the effect of blockage on the numerical results has been investigated systematically on single cylinder cases at low Reynolds numbers [34,35]. For example, Sohankar et al. [35] observed that at the block ratio of 0.025 (H = 40d, where H is the width of the computational domain perpendicular to the oncoming flow), the influence of far field boundary is negligible on the flow near the square cylinder. However, until recently, investigation on the effect of blockage on multiple tandem cylinders is not found in the literatures, so, it is necessary to evaluate the blockage effect on the present flow simulation results. Two different block ratios, respectively of 0.025 (H = 40d) and 0.01 (H = 100d) are selected for the test and the variation of quantities, including mean drag, r.m.s. lift coefficients and Strouhal number for the first, third and the last cylinders at s/d = 5.0 are tabulated in Table 3. In general, the variation trends in flow quantities for all of the selected cylinders are the same, which decrease with the decrease of the block ratio. This may indicate that the effect of block ratio for the multi-cylinder flows is similar with that for single cylinder. As shown in Table 3, on decreasing the block ratio from 0.025 to 0.01, the maximum effect occurred for the r.m.s. lift on the last cylinder, decreasing from 0.163 to 0.157 with variation percentage of 3.77%. The changes in the quantities for other cylinders are less than 3.0%. The maximum variation percentage then shows that the block ratio of 0.025 is acceptable for the present simulation. In the Liang et al. [3], the corresponding block ratio is 0.031, relatively larger than that in the present work. Grid independence study is also carried out with two different meshes (coarser and finer) for the case at the spacing ratio s/d = 5.0. The grid nodes distributed uniformly along the surface of the cylinder are 160 and 200, respectively. The results of the mean drag coefficient, r.m.s. lift coefficient and the Strouhal number obtained for the three upstream cylinders are summarized in Table 4. It is noted that the maximum difference is 3.62% in the

value of mean drag coefficient for cylinder 3, which is acceptable. So the results presented later are all based on the coarser grid system with 160 nodes on the cylinder surfaces. The time step size used for all numerical simulations takes the value of 0.005. 4.1. Flow patterns The numerical results show that the gap spacing has great impact on the flow patterns around inline square cylinder array. The flow mode transition takes place from steady to unsteady wake as the spacing crosses a critical value. As the cylinders are placed sufficiently close, the wake mode is symmetric, and the flow interference is mainly dominated by the proximity effect. Typical flow pattern is shown in Fig. 7 in terms of instantaneous streamlines. As shown in Fig. 7a, the flow is steady for s/d = 1.5, where flow stream passes from the upper and lower sides of each cylinder in symmetric fashion and no vortex shedding occurs in the gap, but stagnant symmetric recirculation zones develop behind each cylinder. With the increase of the spacing ratio up to 3.0 (see Fig. 7b), the flow still remains to be steady, whereas the center of symmetric recirculation zones in each gap moves toward inside slightly due to the increasing driven effect from the free-stream flow. As the spacing varies in the range of 3.5 6 s/d 6 4.5, the flow changes from symmetric to asymmetric patterns; therefore, an important change takes place in the flow feature around this spacing range. The flow becomes no longer steady as evidenced in Fig. 8. At s/d = 3.5 (Fig. 8a), the vortex shedding is not fully developed yet in the gap between cylinder 1 and 2, whereas in the downstream gaps, there are distinct vortices shedding simultaneously from opposite sides of the cylinders. Furthermore, a single row of vortex street is formed in the wake behind the last cylinder. As the spacing is increased to s/d = 4.0, the vortex strength behind the last cylinder is reduced considerably, therefore, a vortex street in two-row configuration is developed in the last wake. As the spacing further increases to s/d = 4.5, the reducing of vortex strength propagates upstream to include last three cylinders, which leads to the wake consisting of two rows of vortices shift toward more upstream. Two representative vorticity contours respectively at the spacings of s/d = 7.0 and 13.0, beyond the critical range for the wake mode transition, are shown in Fig. 9. At s/d = 7.0, as observed in Fig. 9a, two rows of vortices form immediately behind the upstream cylinder 2, however, the vortex shedding in strengthened fashion is recovered in the wake of the last two cylinders. As the spacing further increases to s/d = 13.0, behind each cylinder, a clear vortex street is developed with different wake width. From the above observation, it is summarized that with the increase of the gap spacing, a total of six kinds of wake patterns appear in the flow. They are respectively steady wake (s/d = 1.5, 3.0), non-fully developed vortex street in single row (s/d = 3.5) and double-row (s/d = 4.0), fully developed vortex street in doublerow (s/d = 4.5), fully developed vortex street in partially recovered single-row (s/d = 7.0) and fully developed multiple vortex streets (s/d = 13.0). 4.2. Pressure distributions

Fig. 6. The computational mesh for flow past an inline square cylinder array with s/d = 5.0.

The quantitative analysis on the features of pressure distribution will improve the understanding of flow characteristics of such complicated multi-body interference. In this section, we present the numerical results for the distribution of pressure coefficients in terms of both mean and r.m.s. values along each of the cylinder surface. The distributions of mean pressure coefficient along the cylinder surfaces are shown in Fig. 10. For each cylinder, the numbers 0(4), 1, 2 and 3 represent four different vertices of the square,

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Y. Bao et al. / Computers & Fluids 55 (2012) 118–131 Table 3 Comparison of the global parameters for the block ratio independence study. Block ratio (H/d)

Cylinder 1

0.025 0.01 Error

Cylinder 3

CD

C 0L

St

1.388 1.366 1.54%

0.321 0.314 2.11%

0.122 0.120 1.6%

Cylinder 6

CD

C 0L

St

CD

C 0L

St

0.133 0.130 2.04%

0.548 0.532 2.89%

0.122 0.120 1.6%

0.216 0.212 2.01%

0.163 0.157 3.77%

0.122 0.120 1.6%

Table 4 Comparison of the global parameters for the grid independence study. Grid

Coarser Finer Error

Cylinder 1

Cylinder 2

Cylinder 3

CD

C 0L

St

CD

C 0L

St

CD

C 0L

St

1.388 1.385 0.22%

0.321 0.322 0.31%

0.122 0.122 0

0.549 0.553 0.72%

1.225 1.233 0.65%

0.122 0.122 0

0.133 0.138 3.62%

0.548 0.554 1.08%

0.122 0.122 0

(a) s/d=1.5

(a) s/d=3.5

(b) s/d=3.0

(b) s/d=4.0

Fig. 7. Instantaneous streamlines for flow past inline square cylinder array before the critical gap spacing range.

and are also marked on the horizontal axis, which represents the distance of a specified point on the surface from the starting point of ‘0’ in clockwise direction. The most upstream cylinder 1 shows a similar distribution profile with the case of single square cylinder, as shown in Fig. 10a. For all cases of different spacings, the front surface experiences positive mean pressure, whereas the side and rear surfaces of the cylinder are sucked in negative pressure, indicating that multi-body interference has insignificant impact on the mean pressure around the cylinder 1. A distinctly different distribution is observed for rear five cylinders from Fig. 10b–f. For the front surface of these cylinders, the stagnation pressure takes the value of positive or negative, depending on the spacing distance. As cylinder spacing is very small, the pressure on the front surface takes negative value, due to the fact that in each gap region, there is no vortex shedding, instead, a stagnant symmetric vortex pair is formed. With the increase of spacing width, the flow in the gap becomes unsteady, and then leads to the increase of the pressure, changing from negative to positive values. However, even at s/ d = 14.0, the peak value is only half of the stagnant pressure for single square cylinder, implying the shielding effect of the front cylin-

(c) s/d=4.5 Fig. 8. Instantaneous vorticity contours for flow past inline square cylinder array in the critical gap spacing range.

der still exists at this spacing distance. Both of the side and rear surfaces of downstream cylinders (2–6) show a similar pressure distribution profiles along the perimeter for different cylinders. The absolute values of the pressure on side surfaces of the cylinders are significantly reduced, due to the fact that the Bernoulli effect becomes weak for the rear cylinders, and this trend is strengthened with the decreasing of the gap spacing. On the other hand, the mean pressure on the rear sides is dependent on not only gap spacing but also the cylinder row in the array. For example, for

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backward along the cylinder surface. From Fig. 11, it can also be observed that in the vortex shedding regime, as similar to mean surface pressure, the fluctuating strength of pressure distribution is also depending on both the row position and the spacing distance. For different cylinder, the variation trend of pressure profile with the spacing is different, indicating that the flow interference effect from other cylinders are not the same for each cylinder.

(a) s/d=7.0

(b) s/d=13.0 Fig. 9. Instantaneous vorticity contours for flow past inline square cylinder array after the critical gap spacing range.

cylinder 2, the absolute mean pressure is decreased when compared with the case of single cylinder, but for cylinder 3, the pressure is decreased compared to the single cylinder case in the range of the spacings 3.0 6 s/d 6 10.0, but increased as the spacing is relatively larger (s/d = 14.0). For the most downstream cylinder 6, the situation is much complicated. The suction pressure reaches to a peak value at s/d = 3.5, larger than the corresponding value for single cylinder, but it drops suddenly as the spacing width increases, however, with the further increasing s/d, a second and third peaks are observed at s/d = 8.0, 14.0, respectively. This observation implies that the pressure characteristic of cylinder 6 is not simply dependent on whether the flow is steady or not, but is principally caused by the integrated interference from upstream cylinders. The fluctuating pressure distributions in terms of r.m.s. values along cylinder surfaces are shown in Fig. 11. From the figure, it can be observed that the pressure fluctuations show two different distribution modes along all of the cylinder surfaces. As the gap distance is sufficiently small (s/d = 3.0), the flow is steady, so that the pressure fluctuation is suppressed completely along the perimeter, irrespective of the cylinder location in the array. When the flow changes to unsteady, vortex shedding is developed in the gap region of adjacent cylinders, and then leads to the severe fluctuation of the pressure. From the comparison of the distribution for different cylinders, it is apparent that the distribution mode transition for rear five cylinders occurs almost simultaneously as the spacing varies in 3.0 6 s/d 6 3.5, whereas it occurs for front cylinder 1 as the spacing is varied in 4.0 6 s/d 6 4.5. This indicates that the wake mode transition for cylinders 2–6 from symmetric vortex pair to unsteady vortex shedding are synchronized, but that for the upstream cylinder 1 is delayed to larger gap spacing. The synchronized wake mode transitions for rear cylinders are differentiated from the case of circular cylinders as reported by Liang et al. [3]. They reported that the vortex shedding was started from the downstream cylinders and with the increase of the spacing, they propagated upstream one after another in progressive form [3]. On the other hand, the gap distance associated with the mode transition is relatively shorter than that observed in present simulation. The different behaviors for square and circular cylinders imply that the flow around multiple circular cylinders is more susceptible than square cylinders to become unstable, due to the fact that for square cylinder, the separation point is always fixed on the specified vertices, while for circular one, it moves forward and

4.3. Force statistics Force statistics for drag and lift coefficients are presented in this section for the investigation of the effect of gap spacing on the force characteristics in inline square cylinder array. Fig. 12 shows the variation of mean and r.m.s. values of force coefficients as a function of gap spacing for each cylinder. It can be seen in Fig. 12a that, there are three different kinds of variation trend in mean drag coefficient with gap spacing. The first trend is observed for the upstream cylinder 1, in which the variation-curve of mean drag coefficient ðC D Þ is the most close to C D value for single square cylinder. Furthermore, the curve suddenly jumps up as the spacing increases from s/d = 4.0–4.5, and gradually approaches to the C D of single cylinder with the increase of gap spacing. As the spacing is increased to s/d = 15.0, the C D for cylinder 1 is nearly equal to that for single cylinder, showing that interference from rear cylinders reduces to be negligible. The second kind of variation-curve is evidenced for the second cylinder. Before the fully development of vortex shedding in the upstream gap (s/d 6 4.0), the C D of cylinder 2 takes negative values. As the vortex shedding is fully developed (4.0 6 s/d 6 4.5) C D reverses to positive and jumps up strikingly to a high level, after then it rises gradually with the increasing in gap spacing. However, even at the largest spacing of s/d = 15.0, the C D is nearly 56% of that for single cylinder. When compared to the C D curves for other rear cylinders, it can be implied that the shielding effect from cylinder 1 is most significant on the second cylinder. The third kind of variation trend is observed for downstream cylinders 3–6, which is characterized by multiple peaks of C D with the variation of gap spacing. The first peak for these cylinders occurs when the spacing is ranged in 3.5 6 s/ d 6 4.0, which is associated with the wake mode transition in the gap between successive cylinders. But, the gap distance at which the second peak occurs is different for different cylinders. In fact, as the cylinder in the array is positioned more upstream, the occurrence of the second peak is more delayed. This can be supported by the fact that the second peaks for cylinders 3–6 are respectively correspond to the spacings s/d = 15.0, 11.0, 9.0 and 8.0. Another important feature for rear cylinders 3–6 shown in Fig. 12a is that C D varies in periodic fashion with the increase of gap distance; however, the spacing period for the peak occurrence seems to decrease as the cylinder is positioned more downstream. For example, the third peak for C D of cylinder 6 is available at s/d = 14.0, since the simulated spacing is restricted to s/d 6 15.0, the third peaks for other cylinders are out of sight, but the periodic trend is visible for the cylinders 4 and 5. The variation of the r.m.s. value of drag coefficient fluctuation ðC 0D Þ for each cylinder is shown in Fig. 12b. It is clearly seen that the variation of C 0D also displays three different trends respectively corresponding to the first cylinder, second cylinder and the other four cylinders of 3–6. The C 0D curve for the first cylinder collapses on the value of single cylinder with a small peak at s/d = 4.5, whereas the curve for the second cylinder shows a relatively larger peak at the same spacing, after then, it gradually increases with the increase of the spacing. The C 0D curves for rear four cylinders, including 3–6, display multiple peaks and periodic variation with the increase of the gap spacing, showing a similar trend with their counterpart in terms of mean drag coefficient.

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1.5

0.5 0.0 -0.5 -1.0

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=8.0 s/d=14.0 single

1.0 0.5 0.0

Cp

Cp

1.5

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=8.0 s/d=14.0 single

1.0

-0.5 -1.0

-1.5

-1.5

-2.0

-2.0

-2.5

-2.5

0

1

2

3

4

0

1

(a) cylinder 1

4

0.5 0.0 -0.5 -1.0

1.5

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=8.0 s/d=10.0 s/d=14.0 single

1.0 0.5 0.0

Cp

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=8.0 s/d=10.0 s/d=14.0 single

1.0

Cp

3

(b) cylinder 2

1.5

-0.5 -1.0

-1.5

-1.5

-2.0

-2.0

-2.5

-2.5

0

1

2

3

4

0

1

(c) cylinder 3

2

3

4

(d) cylinder 4 s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=9.0 s/d=10.0 s/d=14.0 single

1.0 0.5 0.0 -0.5

1.5

0.5 0.0 -0.5

-1.0

-1.0

-1.5

-1.5

-2.0

-2.0

-2.5

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=8.0 s/d=10.0 s/d=14.0 single

1.0

Cp

1.5

Cp

2

-2.5

0

1

2

3

4

(e) cylinder 5

0

1

2

3

4

(f) cylinder 6

Fig. 10. Mean surface pressure distribution for each cylinder with different spacing ratios.

As shown in Fig. 12c, the variations of the r.m.s. value of lift fluctuation ðC 0L Þ for six cylinders can also be categorized into three different kinds as similar with those of drag force statistics. The first cylinder experiences a jumping lift fluctuation at the same spacing with that for drag force counterparts, after then, it slowly approaches to the C 0L for single cylinder. The second cylinder undergoes a dramatic jump with the value of C 0L ¼ 1:22 at the same gap spacing with that for the first cylinder, which is the maximum in all of cylinders. With the further increase of gap width, C 0L of the second cylinder decreases gradually, but is maintaining a higher level of value even at the largest gap spacing of s/d = 15.0. The lift fluctuation on the cylinders 3–6 shows a first peak at the gap spacing where wake

mode changes to the vortex shedding in the corresponding gap, after then, a periodic C 0L with the increasing s/d is observed for these cylinders, as similar to the above observations in drag statistics. In the above observation, the wake mode transition from vortex attachment to vortex shedding in the gap is the cause of the sudden first peak at the relatively small gap width. As the gap spacing is larger than that critical spacing, a vortex shedding is fully developed behind each cylinder. However, the periodic variation of force statistics in this regime indicates that the same wake interference effect is periodically appeared with the variation of the gap spacing. The force statistics of inline circular cylinder array also show a similar variation profile at the spacing around the wake mode

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1.0

0.25

'

0.15

0.6 '

Cp

0.20

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=8.0 s/d=14.0 single

0.8

Cp

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=8.0 s/d=14.0 single

0.30

0.4

0.10

0.2

0.05 0.00

0.0 0

1

2

3

0

4

1

(a) cylinder 1

3

4

(b) cylinder 2

0.6

'

0.4

0.9

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=9.0 s/d=10.0 s/d=14.0 single

0.8 0.7 0.6 0.5 '

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=8.0 s/d=10.0 s/d=14.0 single

Cp

0.8

Cp

2

0.4 0.3

0.2

0.2 0.1

0.0

0.0 0

1

2

3

0

4

1

(c) cylinder 3

3

4

(d) cylinder 4

0.7 0.6 '

0.5 0.4

1.0

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=8.0 s/d=10.0 s/d=14.0 single

0.8 0.6 '

s/d=3.0 s/d=3.5 s/d=4.0 s/d=4.5 s/d=9.0 s/d=10.0 s/d=14.0 single

0.8

Cp

0.9

Cp

2

0.4

0.3 0.2

0.2

0.1 0.0

0.0 0

1

2

3

4

(e) cylinder 5

0

1

2

3

4

(f) cylinder 6

Fig. 11. R.m.s. surface pressure distribution for each cylinder with different spacing ratios.

transition with the present numerical results [3]. However, since the simulated gap spacing is restricted to 2.1 6 s/d 6 4.0 in [3], the variation feature for lager gap width is not reported.

4.4. Wake oscillation frequency (Strouhal number) The frequency characteristics of wake oscillation for the flow past multi-cylinder array are analyzed through Fourier transformation of time series for lift force coefficient. As flow interference exists, the lift force series may contain multiple frequency components, which are characterized by multiple frequency peaks on the power spectrum density plot. In the present investigation, the sec-

ondary frequency is considered only if it is of the same order with the dominant frequency. Fig. 13 shows the variation of the vortex shedding frequencies with gap spacing for each square cylinder, along with that for a single square cylinder at the same Reynolds number. It can be seen from Fig. 13a that, both of the first and second cylinders show a single dominant frequency in both flow regimes of vortex attachment (s/d 6 4.0) and vortex shedding (4.5 6 s/d 6 15.0). With the transition from vortex attachment to vortex shedding, the lift frequency is increased and gradually closes to the vortex shedding frequency of single cylinder. A different feature is noticed in Fig. 13b for the third and fourth cylinders. As the gap flow is steady, the lift frequency characteristics for these two cylinders are the

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(a) 1.6

0.15

1.4

St

1.2 1.0

CD

0.6 0.4 0.2

cylinder 1 cylinder 2 single cylinder

0.05

Cylinder 1 Cylinder 2 Cylinder 3 Cylinder 4 Cylinder 5 Cylinder 6 Single

0.8

0.10

3

4

5

6

7

8

9 10 11 12 13 14 15

s /d

(a) cylinder 1 and 2

0.0 -0.2

0.15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

St

s /d

cylinder 3 cylinder 3 (sec) cylinder 4 cylinder 4 (sec) single cylinder

0.10 0.05

(b) 0.5

3

4

5

6

7

8

0.4

(b) cylinder 3 and 4

Cylinder 1 Cylinder 2 Cylinder 3 Cylinder 4 Cylinder 5 Cylinder 6 Single

0.3

C'D

9 10 11 12 13 14 15

s /d

0.2

cylinder 5 cylinder 5 (sec) cylinder 6 cylinder 6 (sec) single cylinder

0.10

St

0.1

0.15

0.05

0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3

s /d

5

6

7

8

9 10 11 12 13 14 15

s /d

(c) 1.4

(c) cylinder 5 and 6

1.2

Fig. 13. Variation of Stouhal number with spacing ratio.

1.0 Cylinder 1 Cylinder 2 Cylinder 3 Cylinder 4 Cylinder 5 Cylinder 6 Single

0.8

C'L

4

0.6 0.4 0.2 0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

s /d Fig. 12. Variation of force coefficients with spacing ratio for each cylinder: (a) mean drag coefficient; (b) r.m.s. value of drag coefficient fluctuation; (c) r.m.s. value of lift coefficient fluctuation.

same with that for the upstream two cylinders of 1 and 2, however, as the flow becomes unsteady with vortex shedding, a secondary frequency begins to appear. In the plot, the ‘up-triangle’ and ‘plus’ symbols denote the secondary frequency respectively for cylinder 3 and 4. Furthermore, in the vortex shedding regime, there are two kinds of frequency distribution according to the gap spacing. As the spacing is ranged in 5.5 6 s/d 6 8.0, the dominant frequency closes to the St of single cylinder while the secondary frequency is nearly half of the dominant one, which indicates that the wake oscillations are dominated by the individual vortex shedding mechanism, but the unsynchronized wake interference effects play only a supplementary role. In contrast, as the spacing is increased up to s/d P 9.0, the wake interference effect reverses to dominate the wake frequency. This is supported by the fact that, in this regime, the secondary frequency is very close to the vortex shedding frequency of single cylinder, while the dominant frequency becomes almost half of the secondary one. This feature is more obvious for the third cylinder. As the flow is steady (s/d 6 4.0), the fifth and sixth cylinders also show the same frequency characteristics with their upstream counterparts. In the vortex shedding regime, the flow interference

effect prevails over the vortex shedding mechanism in the wake oscillation. It is evidenced by the fact that, none of frequency component close to vortex shedding frequency of single cylinder is detected in the power spectrum density plot. As the gap spacing varies in 6.0 6 s/d 6 10.0, only a single dominated frequency is detected for both of two cylinders and equal to each other. When the gap spacing is large enough, a secondary frequency appears again, and with the further increase of the spacing, the dominant frequency for both cylinders reduces to very low value, approximately 23% of that for single cylinder (s/d P 14.0). This observation indicates that multiple wake interference from the upstream cylinders would significantly elongate the predominant wake oscillation period for the most downstream cylinders. From the above discussion, the characteristic of wake oscillation frequency may be summarized as below. In the steady flow regime, all of the cylinders display the same frequency characteristics; however, in the vortex shedding regime, the frequency of wake oscillation is dependent on cylinder location. In general, if the cylinders are located at upstream position of the array, e.g., 1 and 2, the frequency is close to the vortex shedding frequency of single cylinder; if the cylinders are located at middle position of the array, e.g., 3 and 4, multiple frequencies involve in the wake oscillation, but the dominant frequency would jumps down when the spacing is sufficiently large; if the cylinders are located at downstream poison, e.g., 5 and 6, the dominant frequency is considerably small even at relatively small gap spacing, as the spacing is large enough, the frequency would be further dropped.

4.5. Analysis of the instantaneous flow fields for s/d = 3.5 and 4.5 As observed in the previous sections, with the transition of wake mode from symmetric steady flow to asymmetric vortex shedding flow, the characteristics of force statistics change dramatically in response. Furthermore, even in the same wake mode

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of vortex shedding, as the gap spacing varies, the rows of cylinder behave quite distinctly in terms of aerodynamic characteristics. In this section, we choose the cases of s/d = 3.5 and 4.5 to carry out a comprehensive analysis of the instantaneous flow fields in terms of streamlines, pressure and vorticity contours to reveal the flow mechanism in the inline multiple cylinder configurations. Fig. 14 shows the time histories of lift coefficients for s/d = 3.5 and 4.5 in a vortex shedding period. Nearly anti-phase lift fluctuations are observed for adjacent two cylinders as shown in Fig. 14a. It is also seen in the figure that, due to the oscillated recirculation zone in the gap, the fluctuation amplitude for the lift force on cylinder 1 is very weak at this spacing. This is confirmed by looking at the instantaneous streamlines plotted in Fig. 15a. The lift fluctuation is strengthened at the second cylinder and maximized for the last three cylinders. In Fig. 15, the instantaneous flow fields at four time instants are presented, which are denoted by letters A–D in the lift history in Fig. 14a. From the streamline evolutions, it can be observed that the vortices are shed alternatively from the opposite sides of two adjacent cylinders, which explains the antiphase relationship between lift fluctuations of successive cylinders. Let us focus our attention now on the mechanism for the larger amplitude of lift force for the downstream three cylinders. Be correlated with the alternative location for vortex formation, in two successive gaps the direction of the streamlines crossing the gap changes alternatively from top to bottom or vise versa, which illustrates that the wake evolution is synchronized with the cylinder spacing. At this spacing, the flow stream crossing the gap is accelerated due to the ‘jet’ effect generated between the upstream vortex forming zone and immediate downstream cylinder, therefore, leads to the shifting of frontal angle of pushing pressure and the rising of sucking pressure at its opposite side (see Fig. 15b). This then can explain the higher lift fluctuation for the last three cylinders. This mechanism is manifested in the vorticity snapshots (see A

C

B

D

1.5 1.0

cylinder 1 cylinder 2 cylinder 3 cylinder 4 cylinder 5 cylinder 6

CL

0.5 0.0 -0.5 -1.0

404

406

408

410

412

tU /d

(a) s/d=3.5 2.0

A

C

B

D

1.5 1.0

cylinder 1 cylinder 2 cylinder 3 cylinder 4 cylinder 5 cylinder 6

CL

0.5 0.0 -0.5 -1.0 -1.5 -2.0 402

404

406

408

410

5. Conclusion In the present study, the effect of spacing ratio on flow characteristics for an inline cylinder array consisting of six square cylinders is studied numerically at a Reynolds number of 100. The tests for a single square cylinder and two tandem square cylinders are conducted first to validate the code and serve as a reference for further study and the results agree well with other available numerical and experimental results. Then, the simulations for the flow over an inline square cylinder array were carried out on the cylinder spacings, ranging in s/d = 1.5–15.0. Main finding are summarized as follows:

-1.5 402

Fig. 15c) by the clear wake evolution process of vortex formation, vortex shedding and vortex impingement in the gap region. As the spacing increases to s/d = 4.5, vortex shedding is fully developed behind all of the cylinders, including the first one, therefore, the lift force histories for all cylinders vary correspondingly. As shown in Fig. 14b, the lift force amplitude for the first three cylinders is increased due to the vortex impingement effect; in contrast, however that for the last three decreases obviously. It can also be seen in the figure that, the phase angles of lift histories for the first three cylinders are relatively small (close to in-phase), however, the lift variation for the last three cylinders indicates considerable phase difference in the wake evolution. It is illustrated in the instantaneous streamlines (see Fig. 16a) by the fact that the vortices behind the first three cylinders are shed almost simultaneously from the same side, but for the last three, there exists time delay in the vortex shedding. This then implies that the synchronization of vortex motion and gap spacing is not available at this spacing. More importantly, due to the increase of the gap width, the accelerated flow stream crossing the gap is unavailable; instead the flow stream makes a ‘U’ turn within the gap. This is a leading cause for the significant reduce of the lift amplitude for the last three cylinders. It is also evidenced in the pressure contours around the last three cylinders, see Fig. 16b. As observed in Fig. 16c, the vorticity strength in the wake of the last three cylinders is quite weak; consequently, the downstream vortices with reduced vorticity are distributed along two sides of the cylinder array. Liang et al. [3] presented a comprehensive analysis on the instantaneous flow fields around six circular cylinder array at s/d = 3.6 and 4.0. In their study, the flow behaved in a similar way with the present simulation and mechanism behind the flow was also the same with that in the present study. From both of the present and previous simulation [3], it can be summarized that there are two different flow mechanisms responsible for the lift fluctuation jump. The first is the vortex impingement mechanism, which is available for the cylinders located at upstream rows as the spacing is large enough that upstream wake is fully developed; the second is the ‘jet’ flow mechanism, which is applicable for the cylinders in downstream rows when the spacing is appropriately narrow that a ‘jet’ stream from one side to the other can be triggered in the gap.

412

tU /d

(b) s/d=4.5 Fig. 14. Time histories of lift coefficients for each cylinder at different gap spacings.

(i) With the increase of the gap spacing, following six kinds of wake patterns successively appear in the flow for the multiple-cylinder array: steady wake, non-fully developed vortex street in single row and double-row, fully developed vortex street in double-row, fully developed vortex street in partially recovered single-row and fully developed multiple vortex streets. (ii) The mean pressure distribution reveals that the shielding effect on the front surface and the reduction of Bernoulli effect on the side surface of rear cylinders are continuously available in wide range of gap spacing. On the other hand,

Y. Bao et al. / Computers & Fluids 55 (2012) 118–131

(a)

(b)

(c)

Phase A

129

Phase B

Phase C

Phase D

Phase A

Phase B

Phase C

Phase D

Phase A

Phase B

Phase C

Phase D

Fig. 15. Snapshots of instantaneous flow fields at phases A–D (s/d = 3.5): (a) streamlines; (b) pressure contours; (c) vorticity contours.

the wake mode transition from symmetric flow to asymmetric vortex shedding is the main cause of the variation in the mean pressure on rear surface and r.m.s. pressure on the perimeter of each cylinder.

(iii) The first and second cylinders behave similar with two tandem cylinder configuration in terms of aerodynamic force statistics; on the other hand, other four cylinders experience periodic characteristic in force statistics with the increasing of gap spacing.

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(a)

Phase A

Phase C

(b)

(c)

Phase B

Phase D

Phase A

Phase B

Phase C

Phase D

Phase A

Phase B

Phase C

Phase D

Fig. 16. Snapshots of instantaneous flow fields at phases A–D (s/d = 4.5): (a) streamlines; (b) pressure contours; (c) vorticity contours.

(iv) In the steady flow regime, all of the cylinders display the same frequency characteristics; however, in the vortex shedding regime, the frequency of wake oscillation is dependent on cylinder rows. In general, in upstream rows of the array, the wake is dominated by a single frequency

close to the vortex shedding frequency of single cylinder, however, in the downstream rows, there are always multiple frequencies involving in the wake oscillation, and the dominant frequency jumps down as the gap is sufficiently wide.

Y. Bao et al. / Computers & Fluids 55 (2012) 118–131

(v) There exist two different mechanisms for the lift force fluctuation jump for the spacing around the wake mode transition. The first is the vortex impingement mechanism, which is available for the cylinders located at upstream rows as the spacing allows the fully development of upstream wake; the second is the ‘jet’ flow mechanism, which is applicable for the cylinders located at downstream rows when the spacing is appropriately narrow to trigger a ‘jet’ stream from one side of the gap to the other.

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