Three-dimensional numerical simulation on flow past three circular cylinders in an equilateral-triangular arrangement

Ocean Engineering 189 (2019) 106375

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Three-dimensional numerical simulation on flow past three circular cylinders in an equilateral-triangular arrangement Yangyang Gao a, *, Xinchen Qu a, Ming Zhao b, Lizhong Wang a a b

Ocean College, Zhejiang University, Zhoushan, 316021, PR China School of Computing, Engineering and Mathematics, University of Western Sydney, Locked Bag 1797, Penrith, NSW, 2751, Australia

A R T I C L E I N F O

A B S T R A C T

Keywords: Three cylinders Equilateral-triangular configuration Spacing ratio Reynolds number Flow regimes Force coefficient

Flow past three circular cylinders in an equilateral-triangular configuration is simulated for Reynolds numbers 200 � Re � 3900 and spacing ratios 1.25 � L/D � 4.0, where L is the center-to-center distance between each two cylinders and D is the diameter of the cylinder. Five different flow regimes are identified: proximity with deflected flow (P-D), proximity with flip-flop flow (P–F), dual vortex shedding (DV), transition between dual vortex shedding and triple vortex shedding (DV-TV), triple vortex shedding (TV). The boundary spacing ratios between the five regimes are affected by the Reynolds number. With the increase of the spacing ratio or the Reynolds number, the three-dimensionality of the wake flow becomes stronger. Mode B flow pattern with streamwise vortices is found at L/D ¼ 1.25, 3.5 and 4.0 for Re ¼ 1500 and L/D ¼ 1.25 for 3900. The pressure coefficient on the surface of the downstream cylinder C3 is significantly different from that on a single cylinder, leading to significant reduction of the mean drag coefficient on C3 for all the studied spacing ratios and Reynolds numbers. The Strouhal numbers of C1 and C3 are the same and smaller than that of C2 in DV and TV regimes.

1. Introduction Flow around multiple cylinders has been investigated extensively due to its significance both in engineering applications and fundamental fluid mechanics. In many engineering practice, cylindrical structures are often in groups, e.g. groups of chimneys, chemical reaction towers, tension leg offshore platforms and heat exchanger tube. Three cylinders arranged in an equilateral-triangular configuration are widely applied in heat exchangers, cooling systems for nuclear power plants, offshore and ocean engineering (Bouris and Bergeles, 1999; Ozgoren, 2013). A deep understanding of the flow pattern, vortex shedding and fluid force of two and three cylinders in either side-by-side or tandem configuration in flow has been achieved (Zdravkovich, 1977, 1987; Igarashi, 1981; Williamson, 1985; Sumner et al., 1999; Sumner, 2010). However, the studies of complex interference among three cylinders in an equilateral-triangle configuration in flow are rare. If two cylinders are in side-by-side or tandem arrangement in a fluid flow, depending on the distance between the two cylinders, the flow pattern can be classified into three different flow regimes: proximity

interference, wake interference and the combination of the proximity and wake interference (Zdravkovich, 1977, 1987). The three regimes for flow past two tandem cylinder classified by Williamson (1996) are: proximity regime at small gaps where the two cylinders behave as one single body, reattachment regime at intermediate gaps where the shear layers from the upstream cylinder reattach to the surface of the down­ stream cylinder, either permanently or intermittently, and co-shedding regime at large gaps, where vortex shedding occurs from both cylin­ ders. The detailed review of the effects of the Reynolds number and the spacing ratio on the wake flow structures behind two tandem cylinders can be found in the review article by Sumner (2010). It has been well known that the jet flow through the gap between two side-by-side cyl­ inders with a small gap between them biases towards one cylinder, forming a deflected gap flow pattern. If the direction of the deflected gap flow changes from one cylinder to another intermittently, the gap flow is called flip-flop gap flow (Williamson, 1988). The flow patterns, pressure distributions and velocity characteristics for an array of three cylinders in side-by-side and tandem configurations have also been investigated (Guillaume and Larue, 1999; Zhang and

* Corresponding author. E-mail address: [email protected] (Y. Gao). https://doi.org/10.1016/j.oceaneng.2019.106375 Received 7 September 2018; Received in revised form 2 July 2019; Accepted 25 August 2019 Available online 25 September 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. The schematics of the computational domain for three cylinders in an equilateral-triangle arrangement: (a) 3-D view of computation domain; (b) top view of the computation domain; (c) grid mesh around the three cylinders. Table 1 Mesh dependence and model validation for a single cylinder at Re ¼ 100. Mesh Coarse mesh Medium mesh Fine mesh Braza et al. (1986) Jelinek et al. (1995) Rengel et al. (1999)

Table 2 Mesh dependence and model validation for a single cylinder at Re ¼ 3900.

Nodes Circumferential/ spanwise

Node number

CD

St

Mesh

Nodes Circumferential/ spanwise

Node number

CD

C’L

St

100/15 140/20 180/20 – –

180648 289928 330520 – –

1.372 1.368 1.367 1.37 1.36

0.165 0.167 0.166 0.167 0.168

100/15 140/20 180/20 –

331964 517784 740984 –

1.292 1.229 1.22 1.20

0.492 0.544 0.529 –

0.214 0.216 0.216 0.21

0.166

–

1.245

0.541

0.211

–

1.37

–

–

Coarse mesh Medium mesh Fine mesh Matthies and Strang (1979) Joshi and Jaiman (2017) Alessandro et al. (2016) Rajani et al. (2016)

–

–

1.248

0.554

0.19

–

–

1.27

–

0.225

– – –

– – –

1.26 1.098 1.245

– – 0.664

0.207 0.219 0.211

Zhou, 2001; Harichandan and Roy, 2010). Flow past three cylinders in an equilateral-triangle arrangement is more complicated than three cylinders in tandem. Sayers (1987, 1990) carried out experimental studies on flow past three and four cylinders at various spacing ratios. The spacing ratio is defined as L/D, where D and L are the diameter of the cylinders and the center-to-center distance between the cylinders, respectively. The results showed that the flow direction has a significant effect on the force coefficients of the cylinders. Lam and Cheung (1988) investigated flow interference of three cylinders in an equilateral-triangle arrangement at Reynolds numbers of 2.1 � 103 and 3.5 � 103. The bistable flow characteristic is found to be strongly depended on the initial condition. Moreover, if L/D < 4.65 and the flow approaching angle is between 8� and 27� , the vortex shedding behind the upstream cylinder is suppressed by the downstream cylinder. Tat­ suno et al. (1998) found that the effect of flow interference among three cylinders on the pressure on cylinder surfaces is strong when the spacing

Ferre (2017) Pereira et al. (2018)

ratio is small. Gu and Sun (2001) performed an experimental study of flow past three cylinders at Re ¼ 1.4 � 104. It was found that the di­ rection angle of the flow has strong effects on the pressure distribution on the cylinders and four different flow regions are identified with variation of spacing ratios. Pouryoussefi et al. (2011) experimentally investigated the mean force coefficients and Strouhal numbers for three cylinders in an equilateral triangle arrangement at 1.26 � 104 and 6.08 � 104. Numerical studies of flow past three cylinders in an equilateral tri­ angle arrangement were mainly conducted using two-dimensional (2-D) 2

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Fig. 2. Instantaneous iso-surfaces of streamwise vorticityωx andλ2 at Re ¼ 200 at different spacing ratios. The red and blue surfaces denote positive and negative isosurfaces of jωx j ¼ 0:1.jωx j ¼ 0:5at L/D ¼ 1.5,jωx j ¼ 0:00025at L/D ¼ 2.0.Light green surface representsλ2 ¼ 0:1. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 3. Instantaneous iso-surfaces of streamwise vorticityωx andλ2 at Re ¼ 1500 at different spacing ratios. The red and blue surfaces denote positive and negative isosurfaces of jωx j ¼ 1. Only at L/D ¼ 1.25,jωx j ¼ 0:1. Light green surface representsλ2 ¼ 1. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

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Fig. 4. Instantaneous iso-surfaces of streamwise vorticityωx andλ2 at Re ¼ 3900 at different spacing ratios. The red and blue surfaces denote positive and negative iso1. (For interpretation of the references to colour in this figure legend, the surfaces of jωx j ¼ 1. Only at L/D ¼ 1.25,jωx j ¼ 0:1. Light green surface representsλ2 ¼ reader is referred to the Web version of this article.)

numerical models. Bao et al. (2010) conducted a 2-D numerical inves­ tigation on the effects of spacing ratios and incident angles on flow characteristics behind three cylinders in an equilateral-triangular configuration at a low Reynolds number Re ¼ 100. Through numerical simulations at Re ¼ 100 and 200 and various of spacing ratios and incident angles, Zheng et al. (2016) identified three distinct flow pat­ terns. Wu et al. (2006) investigated the effect of lateral channel walls on flow and heat transfer of three circular cylinders in a horizontal channel. The existing experimental studies of flow past three cylinders are mainly focused on the fluid force, pressure distribution and vortex shedding frequency and very limited flow visualization. In addition, the 2-D numerical models are not capable to model the intrinsic threedimensional (3-D) flow characteristics around the cylinders at large Reynolds numbers in the turbulent flow regime. Moreover, little atten­ tion has been paid on the combined effect of spacing ratio and Reynolds number on the three-dimensional flow characteristics behind three cylinders in an equilateral-triangle configuration. The principal aim of the present work is to gain a better under­ standing of the three-dimensional flow characteristics behind three cylinders in an equilateral-triangle configuration (as shown in Fig. 1) through 3-D numerical simulations. Two of the three cylinders (C1 and C3) are in a tandem arrangement and the third cylinder (C3) is on their side. It is expected that the flow has the combined characteristics of flow past two side-by-side cylinders and flow past two-tandem cylinders. The classification of the wake flow pattern and the correlation between the wake flow pattern and the forces on the cylinders are discussed in detail.

center of the three-cylinder system and its x-axis in the incoming flow direction as shown in Fig. 1. Direct numerical simulation (DNS) is adopted to simulate flow past three cylinders at a low Reynolds number of Re ¼ 200 and the 3-D Reynolds-Averaged Navier-Stokes (RANS) model is used to simulate the flow at large Reynolds numbers of Re ¼ 1500 and 3900. DNS is used for Re ¼ 200, the critical Reynolds number for wake flow transitioning from 2-D to 3-D (Zhao et al., 2013), because obtaining accurate numerical results using DNS for such low Reynolds number is achievable. The Shear Stress Transport (SST) k-ω turbulent model is used to simulate the turbulence. The SST k-ω model is found to be suitable to simulate complex boundary layer flows with adverse pressure gradient and separation (Menter et al., 2003; Tom­ boulides et al., 2018). The 3-D RANS equations for conservation of mass and momentum can be written as

∂ui ¼0 ∂xi � ∂ui ∂ uu ¼ þ ∂t ∂xj i j

(1) � 1 ∂p 1 ∂ ∂u þ μ i ρ ∂xi ρ ∂xj ∂xj

�

ρu’i u’j

(2)

where x1 ¼ x, x2 ¼ y and x3 ¼ z, ui is the velocity component in thexi direction,gi is the gravitational acceleration component in thexi -direc­ tion, ρ, μ, t and p are the density of the fluid, fluid dynamic viscosity,

time and pressure, respectively. u’i u’j is the Reynolds stress tensor. The

RANS equations are solved by the Open Source Field Operation and Manipulation (OpenFOAM®). The finite-volume method (FVM) is applied and pressure-velocity coupling is based on the pressure implicit with splitting of operators (PISO) method. The solver pimpleFoam is adopted in the present study. The cell-limited Gauss linear scheme is applied to discretize the convection terms, while the Gauss linear scheme is used to discretize the Laplacian and pressure terms. The CrankNicolson implicit scheme is used for the temporal discretization.

2. Numerical method 2.1. Governing equations A Cartesian coordinate system is defined with its origin located at the

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Fig. 5. Contours of spanwise vorticity on the x-y plane for Re ¼ 200, (a) L/D ¼ 1.25 and (b) L/D ¼ 1.5.

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Fig. 6. Contours of spanwise vorticity on the x-y plane for Re ¼ 200 and L/D ¼ 2.0 to 3.5.

2.2. Boundary conditions

number of 200, 1500 and 3900, respectively. At the inlet boundary, the Neumann condition is used for pressure. The turbulent kinetic energy k and specific dissipation ω at the inlet boundary are given as k ¼

The computational domain shown in Fig. 1 is 30D long in the streamwise direction and 30D in the cross-flow direction (y-direction). The cylinder system is located at the center in the cross flow direction of the domain and 10D downstream the inlet boundary. The computational domain size in the axial direction of the cylinders is about 10D for Re ¼ 200 and 3.14D for Re ¼ 1500 and 3900, which has been proved to be sufficiently long to simulate the three-dimensionality of the wake flow (Patel, 2010). The wake flow for a single isolated cylinder is in Mode A with a spanwise wavelength of between 3D and 4D for Re ¼ 200, and Mode B with a spanwise wavelength of about 1D for Re > 250 (Zhang et al., 1995; Barkley and Henderson, 1996). The length of the cylinders for Re ¼ 200 is longer to ensure the variation of the flow in the spanwise direction can be well captured. It has been found in literature that as L/D > 4.0, the flow in­ terferences among the three cylinders are very weak and each upstream cylinder behaves as an isolated cylinder for both laminar and turbulent flow regimes (Lam and Cheung, 1988; Gu and Sun, 2001; Bao et al., 2010). To identify all possible flow patterns, seven spacing ratios of L/D ¼ 1.25, 1.5, 2.0, 2.5, 3.0, 3.5 and 4.0 are used. The chosen Rey­ nolds numbers cover both laminar and turbulent flow regimes. In addition, they have been used in other studies, making it possible to straightforwardly compare the present results with the published results. The diameter of each cylinder is 0.01 m and the kinematic fluid viscosity is 10 6 m2/s. The inlet flow velocity in the computational domain are set to be 0.02 m/s, 0.15 m/s and 0.39 m/s for Reynolds

1:5ðUIÞ2 and ω ¼ k0:5 =ðC0:25 μ lÞ, respectively, where the turbulence in­ tensity I is 2%, the model constant is Cμ ¼ 0:09 and the turbulence length l is 0.07D (Menter et al., 2003; Patel, 2010; Tian et al., 2013). At the outlet boundary, the pressure is set to zero and the normal gradi­ ents of the velocity, k and ω are zero. On the two lateral boundaries parallel to the flow direction and two end boundaries perpendicular to the cylinder axes, free-slip boundary conditions are applied for velocity and the gradients of k and ω normal to the boundary are zero. On the cylinder surfaces, no-slip boundary is adopted, k is set at zero and ω is calculated as ω ¼ 60ν=ð0:075Δy2 Þ, where Δy is the distance between the first layer of computational nodes and the cylinder surface (Menter, 1994). 2.3. Validation and mesh dependency study To validate the numerical model, numerical simulations of flow past a single cylinder at Reynolds numbers of 100 and 3900 are performed. Three meshes (referred to as coarse, medium and fine meshes, respec­ tively) with different densities are employed. To accurately simulate the boundary layer flow, the non-dimensional distance of the first nodal point to the wall defined as yþ ¼ μf Δy=ν is less than 1, where μf is the friction velocity and Δy is the distance between a node and wall. Com­ parisons between the mean drag coefficient CD , the standard deviation of lift coefficient C’L and the Strouhal number St calculated from different

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Fig. 7. Contours of spanwise vorticity on the x-y plane for Re ¼ 1500 and L/D ¼ 1.25.

meshes with reported numerical results are presented in Tables 1 and 2. The meshes for Re ¼ 3900 are much denser than those for Re ¼ 200 to ensure that the results are converged at such a high Reynolds number. The drag coefficient (CD), the lift coefficient (CL) and the Strouhal number (St) are defined as � � � CD ¼ Fx ρU 2 DH 2 (3) C L ¼ Fy

�

St ¼ fs D=U

� �

ρU 2 DH 2

where Fx and Fy is the fluid force in the x and y-direction, respectively, H is the length of the cylinder, and fs is the frequency of the oscillatory lift force. At Re ¼ 100, the results of CD and St from the medium mesh are different from those from fine mesh by less than 0.3% and 0.2%, respectively. Similarly, at Re ¼ 3900, the differences in CD , C’L and St between the results from medium and fine meshes are smaller than 1%, 5% and 0.5%, respectively. The mean drag coefficient from the medium mesh is 1.368 at Re ¼ 100 and 1.229 at Re ¼ 3900, which falls in the range of published data of 1.36–1.37 for Re ¼ 100 and 1.1 to 1.27 for Re ¼ 3900, respectively. The present Strouhal numbers also agree with other results in Tables 1 and 2 The mesh dependency study shows that

(4) (5)

7

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Fig. 8. Contours of spanwise vorticity on the x-y plane for Re ¼ 1500 and L/D ¼ 2.0.

Fig. 9. Contours of spanwise vorticity on the x-y plane for Re ¼ 1500 and L/D ¼ 2.5 to 4.0.

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flow pattern for a single cylinder starts at Re � 250 and persists until higher Reynolds typically up to about Re ¼ 1000 (Williamson, 1996). The vortex pairs in Mode B flow pattern is found to exist at Re ¼ 1500 but only as L/D ¼ 1.25, 3.5 and 4.0, where well-structured streamwise vortex pairs can be identified in Fig. 3. As L/D ¼ 1.5 to 3.0, streamwise vortex pairs does not exist because strong interaction between cylinders. It is interesting to see that at Re ¼ 3900, well-structured Mode B vortex pairs can still be clearly seen but only when the gap ratio is small (L/D ¼ 1.25). By observing Figs. 2–4, it can be seen that three cylinders with a small gap ratio of L/D ¼ 1.25 delay the transition of the wake flow from two-dimensional to three-dimensional and from Mode B to full turbulent flow. The contours of the spanwise vorticity on the middle section of the cylinder length for different Reynolds numbers are presented in Figs. 5–9. The spanwise vorticity nondimensionalized by ωz ¼ 0 ω z =ðU =DÞ, where the prime stands for the dimensional values. For the convenience of discussion, the gaps between C1 and C2, between C2 and C3 and between C1 and C3 are referred to as G12, G23 and G13, respectively. Generally, the flow through G12 possesses characteristics of flow past two side-by-side cylinders and that through G13 has char­ acteristics of flow past two tandem cylinders. Fig. 5 shows the contours of the spanwise vorticity for Re ¼ 200 and L/D ¼ 1.25 and 1.5. Globally, the three cylinders behave as a single body and there is only one vortex street behind them. The flow through G23 is too weak to have influence on the vortex shedding at L/D ¼ 1.25. A similar flow pattern was observed by Gu and Sun (2001) and Bao et al. (2010), who reported that the gap flow through G23 is always deflected upwards due to the dominate proximity effect at L/D ¼ 1.25. At L/D ¼ 1.5, the shear layers generated through G23 extended further downstream compared with those at L/D ¼ 1.25, but they do not form strong vortices. Because the shear layers from G23 for L/D ¼ 1.5 are stronger than those for L/D ¼ 1.25 and they bias upwards only, the vortices that are shed from the top of C2 are weakened significantly. Fig. 6 shows the contours of the spanwise vorticity on the x-y plane for Re ¼ 200 and L/D ¼ 2.0 to 3.5. The ‘bistable’ flow pattern occurs at small spacing ratios ranging from 1.25 to 2.0. This bistable flow regime is in consistent with that reported by Lam and Cheung (1998), which is 1.27 < L/D < 2.29. As L/D ¼ 2.0 and 2.5, vortex shedding occurs from C2 and C3, forming a dual vortex shedding (DV) pattern. Vortex shed­ ding does not occur from C1 and the flow between C1 and C3 is very similar to the Pattern E flow reported by Igarashi (1981) for two tandem cylinders in flow. The pattern E for two tandem cylinders is character­ ized by one shear layer from the upstream cylinder roll up intermittently in front of the downstream cylinder as shown in the sketch of Fig. 6 (e). In Fig. 6 the shear layer from the top of the upstream C1 always roll up and form a vortex as L/D ¼ 2.0 and 2.5. Since the shear layer from the top of C1 does not contribute the vortex shedding from C1, the vortices from top of C3 is weak. The vortices from the bottom of C3 are very strong because the shear layer from bottom of C1 reattaches onto C3. As L/D ¼ 3.0 and 3.5, vortex shedding from all the cylinders occurs and the flow pattern is defined as triple vortex shedding (TV) pattern. Combi­ nation of vortices in the wake of the cylinders occurs as L/D ¼ 3.0. Similar to that reported by Bao et al. (2010), as the spacing ratio is large enough, the downstream C3 fails to suppress the vortex shedding from C1 and the vortex shedding from downstream cylinder 3 is strongly disturbed by the impingement of vortices shed from upstream C1. Fig. 7 shows contours of the spanwise vorticity on the x-y plane for Re ¼ 1500 and L/D ¼ 1.25. Similar to that for Re ¼ 200, the flow in Fig. 7 is in proximity regime. However, the behavior of the flow through G23 for Re ¼ 1500 and L/D ¼ 1.25 is distinctly different from that for Re ¼ 200 and L/D ¼ 1.25. As Re ¼ 1500 and L/D ¼ 1.25, the direction of the flow through G23 is found to flip up and down frequently. The changeover phenomenon of the deflected directions in G23 is also observed by Tatsuno et al. (1998). However, the up-and-down flipping is found to be random, instead of periodic. For example, the flow through

Table 3 Identified flow regimes for three cylinders. L/D

1.25

1.5

2.0

2.5

3.0

3.5

Re ¼ 200 Re ¼ 1500 Re ¼ 3900

P-D P–F P–F

P-D P–F P–F

DV DV-TV TV

DV TV TV

TV TV TV

TV TV TV

P-D, Proximity with deflected flow; P–F, Proximity with flip-flop flow; DV, Dual vortex shedding; TV, Triple vortex shedding; DV-TV, transition between DV and TV with two and half pair of vortices shed in one vortex shedding period.

Fig. 10. Definition of the stagnation point and separation angles.

the medium mesh density is dense enough to obtain the reliable results for the simulations of flow past three cylinders in an equilateral triangle arrangement as is used in the rest of the simulations. 3. Numerical results 3.1. Wake flow pattern The second negative eigenvalueλ2 of the tensor S2 þ Ω2 is used to visualize the three-dimensional vortex flow pattern, where S and Ω represent the symmetric and antisymmetric parts of the velocitygradient tensor, respectively (Jeong and Hussain, 1995). Figs. 2–4 show the iso-surfaces of non-dimensional streamwise vorticity ωx and λ2 for Re ¼ 200, 1500 and 3900, respectively. The vorticity and eigenvalue

are nondimensionalized by ωx ¼ ω x =ðU =DÞ and λ2 ¼ λ 2 =ðU=DÞ2 , respectively, where the prime stands for the dimensional values. In the following discussion, all the vorticities and eigenvalues are non-dimensional. Re ¼ 200 is the critical Reynolds number for the flow past a single, isolated cylinder transitioning from 2-D to 3-D (Zhao et al., 2013). When the Reynolds number is slightly greater than the critical Reynolds number, flow in the wake of a single cylinder becomes 3-D in mode A and the wavelength of the 3-D feature is about 3–4 diameters (Zhang et al., 1995; Barkley and Henderson, 1996). Numerical simulations at Re ¼ 200 are conducted for a cylinder length of H/D ¼ 10 and it is found that the wake flow is 2-D if L/D ¼ 1.25 (Fig. 2 (a)) and 3-D if L/D � 1.5 (Fig. 2 (b)–(g)). However, the three-dimensionality is very weak at L/D ¼ 2.0, the critical spacing ratio where the flow regime changes. As Re ¼ 1500, the wake flow becomes fully 3-D. The flow in the wake of the three cylinders for Re ¼ 1500 and L/D ¼ 1.25, 3.5 and 4.0 are very similar to the Mode B wake flow reported by Williamson (1988) and Carmo et al. (2010a,b), which is characterized by strong streamwise vortex pairs spaced by a distance of about one diameter. The Mode B 0

0

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G23 biases downwards at t ¼ T/4. After one period, it biases downwards at t ¼ 5T/4. The flow pattern in Fig. 7 is named as proximity regime with flip-flop gap flow. If the jet flow through G23 is deflected upwards or downwards, it is governed by the vortex generated from the bottom shear layer of C2, which is found to be trapped in an area immediately behind C2 most of the time and occasionally released (for example at t ¼ 3T/4 in Fig. 7). At t ¼ T/4, a positive vortex has fully grown behind C2 and the flow through G23 biases downwards. When this positive vortex dissipates at t ¼ 3T/4, the gap flow through G23 biases upwards. After that, another positive vortex will grow and squeeze the direction of the gap flow through G23 downwards again. The flow for L/D ¼ 1.5 and Re ¼ 1500 is also in the proximity regime with flip-flop gap flow. Fig. 8 shows contours of spanwise vorticity on the x-y plane for Re ¼ 1500 and L/D ¼ 2.0. While vortex shedding from C2 and C3 can be clearly seen in Fig. 8, it is interesting to see that only vortices from bottom of C1 are shed. Once a vortex is shed from the bottom side of C1, it sweeps C3’s bottom surface and is convected downstream as seen in Fig. 8 (b) and (c). Because of the additional vortex shedding from the bottom side of C1, three positive vortices and two negative vortices can be seen right behind the cylinders in Fig. 8 (c) and (d). L/D ¼ 2.0 for Re ¼ 1500, where two and half pairs of vortices are shed from the three cylinders in one vortex shedding period, is essentially the L/D for the flow transitioning from dual vortex shedding to triple vortex shedding regimes. Fig. 9 shows contours of spanwise vorticity on the x-y plane for Re ¼ 1500 and L/D ¼ 2.5 to 4.0. The critical spacing ratio for flow

changing from dual to triple vortex regimes for Re ¼ 1500 is smaller than that for Re ¼ 200. It is between L/D ¼ 2.5 and 3.0 for Re ¼ 200 and about L/D ¼ 2.0 for Re ¼ 1500. In the dual vortex shedding flow pattern, the vortex wake is very different from that of a single cylinder because of the interaction between vortices. Although two pairs of vortices are shed from the cylinder in each vortex shedding period, the wake vortices are not aligned in two rows. Due to the presence of C2, the wake flow pattern behind C1 is asymmetrical to the centerline, which is consistent with the result obtained from Lam and Cheung (1988). In the triple vortex shedding regime (L/D ¼ 2.5, 3.0, 3.5 and 4.0 for Re ¼ 1500), the vortex shedding from C2 is similar to that of a single cylinder and that from C1 and C3 is similar to the vortex shedding from two tandem cylinders. The vortex shedding patterns identified for Re ¼ 200 and 1500 are also observed for Re ¼ 3900. The flow patterns for Re ¼ 3900 are not shown to avoid repeating. However, the boundary L/D between dual and triple vortex shedding regimes for Re ¼ 3900 is between 1.5 and 2.0, which is smaller than that for Re ¼ 1500. And the boundary L/D between the proximity flow regime and dual vortex shedding regime is between 1.25 and 1.5, which is also smaller than that for Re ¼ 1500. Table 3 lists the identified flow regimes for all the simulated cases. The flow around C1 and C3 in the DV and TV regimes is similar to the regimes found for flow past two tandem cylinders. Different researchers classify the wake flow of two tandem cylinders in different ways. The regimes of SG (symmetric in the gap) and AG (alternating in the gap) defined by Carmo et al. (2010a,b) belong to a single wake and WG (wake in the gap) belongs to the dual wake. The proximity regime,

Fig. 11. Time histories of the angle of stagnation point. Left column: Re ¼ 200; Right column: Re ¼ 1500. 10

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Fig. 11. (continued).

reattachment regime and co-shedding regime defined by Papaionannou et al. (2006) is equivalent to the SG, AG and WG, respectively. The boundary L/D between AG and WG for Re ¼ 500 and 1000 was found to be between 3.5 and 4.0 by Papaioannou et al. (2006). The flows past C1 and C3 in the dual vortex shedding and triple vortex shedding regimes are the same as AG and WG. However, the critical between AG and WG in the present three-cylinder case is found to be much smaller than that in previous studies of two tandem cylinders. This is because the exis­ tence of C2 breaking the symmetry of the system, and C2 acts as a disturbance that triggers WG at smaller L/D. The vortex shedding in the P-D and P–F regimes has some similarity

to the deflected flow regime and flip-flop regime for flow past two sideby-side cylinders, respectively, but not exactly the same. In the deflected flow regime in the two side-by-side cylinder case, the flow can be deflected towards one side for very long time but it can switch from one side to another side. In the P-D regime of the three-cylinder case, the flow is always deflected towards C2. In the flip-flop regime of the two side-by-side cylinder case, the frequency of the changing direction of the deflected flow is still much higher than that of the vortex shedding flow. However, in the three-cylinder case, the flow through G23 changes its direction up and down with a period the same as the vortex shedding period, though the change of direction does not synchronize with the 11

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Fig. 12. Variations of front stagnation point with spacing ratio at different Reynolds numbers.

vortex shedding. Significant Reynolds effects are clearly visible from the instanta­ neous iso-surfaces of streamwise vorticity and spanwise vorticity con­ tours shown in Figs. 2–9. With increasing Reynolds numbers, the threedimensionality becomes stronger. At low Reynolds number Re ¼ 200, the vortex tube are almost aligned parallel to the cylinders and only small scales of streamwise vortices are observed. The inclination of the spanwise vortices is not observed due to the synchronized phase of the vortex shedding along the cylinder span. As Re increases to 1500 and 3900, strong rib-shaped streamwise vortices are generated and shed in the wake of the three cylinders. Due to the strong interference of the streamwise vortices in the wake of the three cylinders, the spanwise vortices become wavy in the wake of the cylinders. The inclination of the spanwise vortices occurs due to the phase differences of vortex shedding along the cylinder span. The critical spacing ratios for different flow regimes for flow around three cylinders are also sensitive to Reynolds number. At a given spacing ratio L/D ¼ 2.0, the flow pattern is DV for Re ¼ 200, DV-TV for Re ¼ 1500, and TV for Re ¼ 3900. The sensitivity of the flow patterns to the Reynolds number also exists for flow past two cylinders in tandem and staggered configurations (Igarashi, 1981, 1984; Sumner et al., 2000).

3.2. Stagnation point and separation point The stagnation point of a cylinder is a point on the upstream side of the cylinder surface, where the shear stress is zero and the pressure is at its maximum (Williamson, 1988). Fig. 10 shows the definition of the stagnation point and separation points on a cylinder. The location of the stagnation point is quantified using the angle of the stagnation, which is zero at the most upstream point of the cylinder surface and measured in the clockwise direction. The stagnation point oscillates along the cyl­ inder surface because of the vortex shedding flow. Fig. 11 shows the time histories of the angle stagnation point for Re ¼ 200 and 1500. While the stagnation point of a single cylinder is nearly stationary, stagnation points of the three cylinder oscillates along the cylinder surface with an amplitude between 1� and 2� as L/D ¼ 1.25. The oscillation amplitude of the angle of stagnation point reduces significantly as L/D ¼ 1.5, and it increases again as L/D ¼ 2.5 for Re ¼ 200 and L/D ¼ 2.0 for Re ¼ 1500. In these two cases, the flow between C1 and C3 is in the dynamic AG mode, which leads to strong oscillation of the angle of stagnation point. When vortex shedding from C1 occurs, C3 is affected by the vortices that are shed from C1 significantly and no stagnation point can be identified on C3.

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Fig. 13. Time histories of the separation angles for three cylinders. Left column: Re ¼ 200; Right column: Re ¼ 1500.

Fig. 14. Comparisons of the results between (a) the present study and (b) Ozgoren (2013).

Fig. 12 shows the variation of time-averaged angle of stagnation point with the spacing ratio for three cylinders at different Reynolds numbers. The angle of stagnation point of C3 is the largest, which is slightly greater than 30� as L/D ¼ 1.25 and aligns well with G12. With increasing L/D, the angle of stagnation point of C3 increases. Similar to what reported by Pouryoussefi (2011), the jet flow causes to diminish the vortex shedding from downstream cylinder and leads to the shifting

of separation points. The angle of stagnation point of C3 does not exist for large L/D because the vortex shedding from C1 makes C3 attacked by vortices and sometime surrounded by vortices. The angle of stagnation points of C1 and C2 are in opposite signs and their magnitudes are close to each other at large L/D. It is interesting that the angle of stagnation angle of the most upstream C1 is greater than C2 at the smallest L/D ¼ 1.25. The angles of stagnation point for C1 and C2 are still slightly 13

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Fig. 15. Time-averaged vorticity contours (top row) and streamlines (bottom row) at different spacing ratios for Re ¼ 3900. (a) L/D ¼ 1.5; (b) L/D ¼ 2.0; (c) L/D ¼ 3.0.

deviated from zero at L/D ¼ 4.0, indicating there is still interaction among these two cylinders. Because vortex shedding flow of C3 is affected by both C1 and C2, the flow separation on its surface is found to be very different from that of a single cylinder. Fig. 13 shows the time histories of the separation angle on the cylinder surfaces at small spacing ratios of L/D ¼ 1.25 and 1.5. As shown in Fig. 10, the separation angle is defined as the angle measured along the cylinder surface from the front stagnation point to the sepa­ ration point. θst and θsb represent the separation angles on the top and bottom sides of the cylinder, respectively. The calculated mean sepa­ ration angles of the single cylinder for Re ¼ 200, 1500 and 3900 are θs ¼ 111.9� , 97.9� , 89.9� , respectively, which are in good agreement with the previous research results (Williamson, 1996; Harichandan and Roy, 2012; Zhao et al., 2013; Joshi and Jaiman, 2017). It can be seen that both θst and θsb of C3 oscillate with the highest amplitude compared with those of the other two cylinders, mainly because it is affected the most by the dynamics gap flow through G23 and the strong vortices generated by the shear layer generated by C1 and strengthened when it passes C3.

contours, streamline topologies, root-mean-square streamwise velocity

(urms =U0 ) and Reynolds stress contours (u0 u0 =U20 v0 v0 =U20 u0 v0 =U20 ) are concentrated in the near wake and symmetrical about the wake centerline, the numerical results are in good agreement with the experimental results in Ozgoren (2013). Figs. 15 and 16 present the time-averaged vorticity contours, streamlines and Reynolds stresses behind the three cylinders at different spacing ratios L/D ¼ 1.5, 2.0 and 3.0 for Re ¼ 3900. It can be seen that due to the complex interferences among the three cylinders, the timeaveraged flow features are observed to be more complicated than those of a single cylinder. At L/D ¼ 1.5, the time-averaged streamline patterns illustrate that the reverse flows are obviously observed in the near wake, and also in the gap between C1 and C3. However, due to the gap flow deflection, the two recirculation zones in the wakes of C2 and C3 are aligned slightly towards top right. The recirculation zone in the wake of C1 is very small and biased. Because the strong vortices in the wake of C2 and C3 as discussed above, the recirculation zone between C2 and C3 are very large. As L/D is increased to 2.0, the recirculation zone in the wake of C1 increases and that in the wake of C2 and C3 decreases, mainly because the reattachment of the shear layers on the downstream C2. Similar what was reported by Ozgoren (2013), the

3.3. Time-averaged flow field

Reynolds shear stress (u0 v0 =U20 ) for the equilateral triangle arranged three cylinders are greater than that of the single cylinder. As L/D ¼ 1.5, strong Reynolds stress is found behind G23, as the result of the flipping gap flow. As L/D ¼ 2.0, the distribution of the Reynolds stress behind C2 becomes symmetric and that behind C3 is slightly asymmetric because of

The time-averaged vorticity contours, streamlines and Reynolds stresses behind a single cylinder are compared with the experimental results obtained from Ozgoren (2013) in Fig. 14. Solid lines represent positive values and dashed lines represent negative values in the con­ tours. In the case of an isolated cylinder, the contours of vorticity

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Fig. 16. Reynolds stress contours at different spacing ratios for Re ¼ 3900: (a) u0 u0 =U20 ; (b) v0 v0 =U20 ; (c) u0 v’=U20 .

the DV flow pattern. However, the Reynolds stress between C1 and C3 is still small because there is no vortex shedding from C1. When L/D ¼ 3.0, the flow is in the TV flow pattern and the Reynolds stress distribution in the wake of the three cylinder are more or less symmetric.

greater than those of the other two cylinders for both Re ¼ 200 and 3900, mainly because C3 is strongly affected by the gap flow through G23 at small spacing ratios and by the vortex shedding from C1 when the spacing ratio is large. At spacing ratios L/D ¼ 1.25 and 2.0 and Re ¼ 200, the oscillations of the lift coefficients of all the cylinders are very weak because no vortices are shed from the upstream C1. As L/D is increased to 3.0, the amplitudes of the lift coefficients of the three cylinders become regular and periodic. Unlike those at Re ¼ 200, the drag and lift coefficients for Re ¼ 3900 of the three cylinders are very irregular due to strong three dimensionality of the wake flow behind the three cylinders. The drag coefficient of C3 is the smallest because it is within the wake zone of C1 and its amplitude increases significantly when vortex

3.4. Pressure distribution and force coefficient Figs. 17 and 18 show the time histories of the drag and lift co­ efficients for Reynolds numbers Re ¼ 200 and 3900, respectively. The drag and lift coefficients of all the cylinders for Re ¼ 200 are more pe­ riodic than those at Re ¼ 3900, because the three-dimensionality is weak. The amplitude of the lift coefficient on the downstream C3 is

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Fig. 17. Time histories of drag (left column) and lift (right column) coefficients for the three cylinders at Re ¼ 200. (a) L/D ¼ 1.25; (b) L/D ¼ 2.0; (c) L/D ¼ 3.0.

shedding from C1 occurs in the TV flow pattern. Fig. 19 shows the variation of the mean pressure coefficient Cp on each cylinder with the spacing ratio, where the overbar means averaged p∞ Þ=ð0:5ρU2 Þ, wherep∞ is the value and Cp is defined asCp ¼ ðp pressure of the incoming flow. Figs. 20–22 show the statistic values of the drag and lift coefficient on the three cylinders for Re ¼ 200, 1500 and 3900, respectively. In the following discussion, CDn and CLn are used to stands for the drag and lift coefficient of the n-th cylinder, respectively. As L/D ¼ 1.25, the blockage effect from the narrow gap G12 in­ creases the pressure between 0� <θ < 120� , resulting in a significant in­ crease in the magnitude of the negativeCL1 and a slight increase inCD1 . With the widening of G12, the magnitude of CL1 decreases. CD1 is smaller than CD of a single cylinder as L/D � 2 for Re ¼ 200 and L/D � 1.5 for Re ¼ 1500 and 3900 because the pressure at the back of C1 is increased. CD2 varies with θ with an opposite trend of CD1 . It is smaller than the corresponding single cylinder value at small L/D and greater at large L/

D. CD1 is reduced at large L/D because the base pressure of C1 is increased compared with that of a single cylinder as the result of weakened vortex shedding in G13 compared with that of a single cyl­ inder. The stronger blockage of the three cylinder system than a single cylinder results in a decrease in the base pressure of C2 and an increase of CD2 . Sayers (1987) reported that the increasing of drag coefficient CD2 at large spacing ratios is due to the proximity interference between C2 and C3. The pressure coefficient on C3 is significantly different from that on a single cylinder. Tatsuno et al. (1998) mentioned that the intense gap flow among the three cylinders causes a reduction in pressure in the gap, leading to the decreasing of drag coefficient. The small high pressure region on C3 near the stagnation point (about 30� ) is due to the attack of the high-speed jet flow through G12. Compared with that of a single cylinder the high-pressure region in front of C3 is much smaller, resulting a significant reduction of the mean drag coefficient of C3 as shown in Figs. 14 to 15. The shift of the stagnation point towards the top

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Fig. 18. Time histories of drag (left column) and lift (right column) coefficients for the three cylinders at Re ¼ 3900. (a) L/D ¼ 1.25; (b) L/D ¼ 2.0; (c) L/D ¼ 3.0.

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Fig. 19. Mean pressure coefficient (Cp ) distributions on the surfaces of cylinders at different spacing ratios and Reynolds numbers: (a) Re ¼ 200; (b) Re ¼ 1500; (c) Re ¼ 3900.

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Fig. 20. Variation of force coefficients with spacing ratio at Re ¼ 200: (a) mean drag coefficient; (b) mean lift coefficient; (c) standard deviation of drag coefficient; (d) standard deviation of lift coefficient.

surface of C3 leads to a negative mean lift coefficient of C3. The pressure at the stagnation point and the lift coefficient at L/D ¼ 1.5 is greater than that at L/D ¼ 1.25, because the larger gap at L/D ¼ 1.5 makes larger jet flow rate through C12 that attacks the front surface of C3. As L/D > 1.5, the pressure at the stagnation point of C3 reduces with increasing L/D, so does the magnitude of the mean lift coefficient because the jet ve­ locity in G12 is decreased. It is interesting to see that the strong oscillation of the angle of separation of C3 at small L/D ¼ 1.25 and 1.5 does not produce strong oscillation of the lift coefficient. When L/D is small, the jet flow velocity and flow rate through G23 are too small to have big effects on the forces. Although the shear layer from G23 separates from and reattach to the back of C3 alternatively, making a big oscillation amplitude of angle of separation, the weakness of the shear layer does not contribute much to the oscillation of the lift coefficient. The standard deviation (SD) of the lift coefficient of two side by side cylinders in flow with flip-flop gap flow at small gap ratios was also found to be smaller compared with that of a single cylinder (Thapa et al., 2015).

From Fig. 20 it can be seen that the strong single vortex street in the proximity regime at small L/D does not increases SD of the lift co­ efficients of all the cylinders, mainly because there is not an integral vortex street in each individual cylinder. Based on Fig. 20 (b) and Fig. 21 (b), the mean lift coefficient of the whole cylinder system (i.e. the sum of the mean lift coefficients of all the cylinders) direct downwards espe­ cially when L/D is small, because the shear layer from the bottoms of C1 and C3 generate stronger vortices than the shear layer from the top of only C2. The sum of the mean drag coefficients of the three cylinders is more than three times the drag coefficient of a single cylinder because CD2 is decreased dramatically. Based on Fig. 20 (c) and (d) and Fig. 21 (c) and (d) we can conclude that put three cylinder together in an equilateral-triangle arrangement at L/D between 2.5 and 4.0 will in­ crease the SD lift coefficient of the whole system. The numerical results by Bao et al. (2010) for Re ¼ 100 are also included in Fig. 20 for comparison. The present variation trend of the force coefficients with L/D is in agreement with that in Bao et al. (2010). However, the boundary L/D between different flow patterns and the

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Fig. 21. Variation of force coefficients with spacing ratio at Re ¼ 1500 and 3900: (a) mean drag coefficient; (b) mean lift coefficient; (c) standard deviation of drag coefficient; (d) standard deviation of lift coefficient.

values of the force coefficients are affected by the Reynolds number. The RMS lift coefficient for Re ¼ 200 increases significantly as L/D increases from 2.5 to 3.0, where the flow changes from DV to TV flow pattern. The significant increase of the lift coefficient for Re ¼ 100 occurs at a greater 0 value of L/D in Bao et al. (2010). In addition, the maximum values of C D 0 and C L for Re ¼ 100 are lower than those for Re ¼ 200.

frequency is called Strouhal number which is calculated by St ¼ fsD/U. Fig. 23 shows the variations of the Strouhal number with L/D for Re ¼ 200, 1500 and 3900. In the proximity regime with L/D ¼ 1.25, the peak frequencies of three cylinders are the same, because the oscillation of all the lift coefficients is caused by only vortex street. When L/D ¼ 1.5, the spectra of C1 and C2 are broad banded, mainly because the strong up-and-down flip-flop jet flow through G23 does not synchronize with the vortex shedding as discussed in Section 3.1. However, the local peak with St ¼ 0.122 can still been seen at Re ¼ 3900, which is the result of the vortex shedding. The Strouhal numbers in the proximity regime are nearly half that of a single cylinder because the dimension of the whole three-cylinder system is nearly twice the cylinder diameter, which leads to decrease in the vortex shedding frequency of the single vortex street. When the flow is in the DV shedding and TV flow regimes, the Strouhal

3.5. Strouhal number The vortex shedding frequency is determined through the Fast Fourier Transform (FFT) analysis of the lift coefficient at the mid-section of the cylinders. Fig. 22 shows the power spectra of the lift coefficient of each cylinder at different spacing ratios for Re ¼ 3900. Each spectrum in Fig. 22 has at least one distinct peak frequency. The nondimenional peak

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Fig. 22. Power spectral density (PSD) of the lift coefficient coefficients at Re ¼ 3900: (a) L/D ¼ 1.25; (b) L/D ¼ 1.5; (c) L/D ¼ 2.0; (d) L/D ¼ 2.5; (e) L/D ¼ 3.0; (f) L/D ¼ 4.0.

numbers of C1 and C3 are the same, because of the synchronization of the vortex shedding from these two cylinders (Sumner et al., 2005; Carmo and Meneghini, 2006; Bao et al., 2010; Pouryoussefi et al., 2011). The Strouhal number of C2 is different from those of C1 and C3 because C2 has little interaction with the other two cylinders. Interestingly, at L/D ¼ 2.0, some cylinders have dual frequencies because the vortex street in the wake of C2 and the vortex street in the wake of C1 and C3 interact with each other. As Re ¼ 200, the St of C1 and C3 are still smaller than that of a single cylinder and the St of C2 is greater. When the Reynolds number increases to 1500 and 3900, the St of C1 and C3 approach St of a single cylinder, but the St of C2 is still higher.

number. Three-dimensionality of primary vortex shedding in­ creases with increasing spacing ratio or Reynolds number. Mode B flow pattern with streamwise vortex pairs is found at L/ D ¼ 1.25, 3.5 and 4.0 for Re ¼ 1500 and at L/D ¼ 1.25 for 3900. (2) The downstream C3 has the largest angle of stagnation point. With increasing L/D, the angle of stagnation point of C3 in­ creases, the angles of stagnation point of C1 and C2 are in opposite signs and their magnitudes are close to each other at large L/D. The separation angle of C3 oscillates with the highest amplitude compared with those of the other two cylinders due to the dynamics gap flow through G23 and the strong vortices generated by C1, and strengthened when it passes C3. (3) The pressure coefficient experienced by the downstream cylinder C3 is significantly different from that on a single cylinder, leading to a smaller mean drag coefficient on C3 than those on C1 and C2, irrespective of the spacing ratios and Reynolds numbers. Both the mean lift coefficients experienced by C1 and C3 are almost negative but positive for C2 at all the examined spacing ratios. The strong single vortex street in the proximity regime at small L/ D does not increases SD of the lift coefficient for all the cylinders, but increases the SD of lift coefficients at large spacing ratios 2.5 � L/D � 4.0. (4) The Strouhal numbers corresponding to vortex shedding experi­ enced by C1 and C3 are mostly the same and increase with spacing ratio, then approach to that of a single cylinder at the triple vortex shedding regime. The Strouhal number of C2 is much larger than those of C1 and C3 at all the examined spacing ratios and maintained at a value of 0.23.

4. Conclusions Three-dimensional numerical simulations on flow past three circular cylinders in an equilateral-triangular arrangement are performed using the OpenFOAM® open source code. The combined effects of spacing ratio and Reynolds number on the three-dimensional flow features and force coefficients are investigated. The findings are summarized as follows: (1) Depending on the spacing ratios and Reynolds numbers, five different flow regimes are identified and they are: P-D (proximity with deflected flow), P–F (proximity with flip-flop flow), DV (dual vortex shedding), DV-TV (transition between dual vortex shedding and triple vortex shedding), TV (triple vortex shed­ ding). The critical spacing ratios for the transition among different flow regimes decrease with increasing Reynolds

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Fig. 23. Variation of St with spacing ratios: (a) Re ¼ 200; (b) Re ¼ 1500; (c) Re ¼ 3900.

Acknowledgements

Harichandan, A.B., Roy, A., 2010. Numerical investigation of low Reynolds number flow past two and three circular cylinders using unstructured grid CFR scheme. Int. J. Heat Fluid Flow 31 (2), 154–171. Harichandan, A.B., Roy, A., 2012. Numerical investigation of flow past single and tandem cylindrical bodies in the vicinity of a plane wall. J. Fluids Struct. 33, 19–43. Igarashi, T., 1981. Characteristics of the flow around two circular cylinders arranged in tandem: 1st report. Bull. JSME 24 (188), 323–331. Igarashi, T., 1984. Characteristics of the flow around two circular cylinders arranged in tandem: 2nd report, unique phenomenon at small spacing. Bull. JSME. 27 (233), 2380–2387. Jelinek, J., Herfjord, B.O., Blakset, T.J., Osvoll, H., Morton, D., 1995. The use of cathodic protection current density surveys for optimizing offshore surveys and anode retrofit design. In: NACE international annual conference and corrosion show in United States. Jeong, J., Hussain, F., 1995. On the identification of a vortex. J. Fluid Mech. 285, 69–94. Joshi, V., Jaiman, R.K., 2017. A variationally bounded scheme for delayed detached eddy simulation: application to vortex-induced vibration of offshore riser. Comput. Fluids 157, 84–111. Lam, K., Cheung, W.C., 1988. Phenomena of vortex shedding and flow interference of three cylinders in different equilateral arrangements. J. Fluid Mech. 196 (196), 1–26. Matthies, H., Strang, G., 1979. The solution of nonlinear finite element equations. Int. J. Numer. Methods Eng. 14 (11), 1613–1626. Menter, F.R., 1994. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32, 1598–1605. Menter, F.R., Kuntz, M., Langtry, R., 2003. Ten years of industrial experience with the SST turbulence model. Turbul. Heat Mass Transf. 4 (1), 625–632. Ozgoren, M., 2013. Flow structures around an equilateral triangle arrangement of three spheres. Int. J. Multiph. Flow 53, 54–64. Papaioannou, G.V., Yue, D.K.P., Triantafyllou, M.S., Karniadakis, G.E., 2006. Threedimensionality effects in flow around two tandem cylinders. J. Fluid Mech. 558, 387–413. Patel, Y., 2010. Numerical Investigation of Flow Past a Circular Cylinder and in a Staggered Tube Bundle Using Various Turbulence Models. Master’s thesis. Lappeenranta University of Technology.

This research was supported by the Joint Fund of Zhejiang Provincial Natural Science Foundation (Grant No. LHZ19E090004), National Key R&D Program of China (2018YFE0109500, 2018YFD0900901). References Alessandro, V.D., Montelpare, S., Ricci, R., 2016. Detached-eddy simulations of the flow over a cylinder at Re ¼ 3900 using OpenFOAM. Comput. Fluids 136, 152–169. Bao, Y., Zhou, D., Huang, C., 2010. Numerical simulation of flow over three circular cylinders in equilateral arrangements at low Reynolds number by a second-order characteristic-based split finite element method. Comput. Fluids 39 (5), 882–899. Barkley, D., Henderson, R.D., 1996. Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241. Bouris, D., Bergeles, G., 1999. 2D LES of vortex shedding from a square cylinder. J. Wind Eng. Ind. Aerodyn. 80 (1–2), 31–46. Braza, M., Chassaing, P., Minh, H.H., 1986. Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech. 165, 79–130. Carmo, B.S., Meneghini, J.R., 2006. Numerical investigation of the flow around two circular cylinders in tandem. J. Fluids Struct. 22 (6), 979–988. Carmo, B.S., Meneghini, J.R., Sherwin, S.J., 2010. Possible states in the flow around two circular cylinders in tandem with separations in the vicinity of the drag inversion spacing. Phys. Fluids 22 (5), 054101. Carmo, B.S., Meneghini, J.R., Sherwin, S.J., 2010. Secondary instabilities in the flow around two circular cylinders in tandem. J. Fluid Mech. 644, 395–431. Ferrer, E., 2017. An interior penalty stabilised incompressible discontinuous GalerkinFourier solver for implicit large eddy simulations. J. Comput. Phys. 348, 754–775. Gu, Z., Sun, T., 2001. Classifications of flow pattern on three circular cylinders in equilateral-triangular arrangements. J. Wind Eng. Ind. Aerodyn. 89 (6), 553–568. Guillaume, D.W., Larue, J.C., 1999. Investigation of the flopping regime with two-, threeand four-cylinder arrays. Exp. Fluid 27 (2), 145–156.

22

Y. Gao et al.

Ocean Engineering 189 (2019) 106375 Tian, X.L., Ong, M.C., Yang, J.M., Myrhaug, D., 2013. Unsteady RANS simulations of flow around rectangular cylinders with different aspect ratios. Ocean. Eng. 58, 208–216. Tomboulides, A., Aithal, S.M., Fischer, P.F., Merzari, E., Obabko, A.V., Shaver, D.R., 2018. A novel numerical treatment of the near-wall regions in the k-ω class of RANS models. Int. J. Heat Fluid Flow 72, 186–199. Williamson, C.H.K., 1985. Evolution of a single wake behind a pair of bluff bodies. J. Fluid Mech. 159 (159), 1–18. Williamson, C.H.K., 1988. The existence of two stages in the transition to threedimensionality of a cylinder wake. Phys. Fluids 31 (11), 3165–3168. Williamson, C.H.K., 1996. Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28 (1), 477–539. Wu, H.W., Perng, S.W., Huang, S.Y., Jue, T.C., 2006. Transient mixed convective heat transfer predictions around three heated cylinders in a horizontal channel. Int. J. Numer. Method H. 16 (6), 674–692. Zdravkovich, M.M., 1977. Review of flow interference between two circular cylinders in various arrangements. J. Fluids Eng. 99 (4), 618–633. Zdravkovich, M.M., 1987. The effects of interference between circular cylinders in cross flow. J. Fluids Struct. 1 (2), 239–261. Zhang, H.J., Zhou, Y., 2001. Effect of unequal cylinder spacing on vortex streets behind three side-by-side cylinders. Phys. Fluids 13 (12), 3675–3686. Zhang, H.Q., Fey, U., Noack, B.R., K€ onig, M., Eckelmann, H., 1995. On the transition of the cylinder wake. Phys. Fluids 7 (4), 779–794. Zhao, M., Thapa, J., Cheng, L., Zhou, T., 2013. Three-dimensional transition of vortex shedding flow around a circular cylinder at right and oblique attacks. Phys. Fluids 25 (1), 014105. Zheng, S.L., Zhang, W., Lv, X.C., 2016. Numerical simulation of cross-flow around three equal diameter cylinders in an equilateral-triangular configuration at low Reynolds numbers. Comput. Fluids 130, 94–108.

Pereira, F.S., Vaz, G., Eça, L., Girimaji, S.S., 2018. Simulation of the flow around a circular cylinder at Re¼ 3900 with Partially-Averaged Navier-Stokes equations. Int. J. Heat Fluid Flow 69, 234–246. Pouryoussefi, S.G., Mirzaei, M., Pouryoussefi, S.M., 2011. Force coefficients and Strouhal numbers of three circular cylinders subjected to a cross-flow. Arch. Appl. Mech. 81 (11), 1725–1741. Rajani, B.N., Kandasamy, A., Majumdar, S., 2016. LES of flow past circular cylinder at Re ¼ 3900. J. Appl. Fluid Mech. 9 (3), 1421–1435. Rengel, J.E., Sphaier, S.H., 1999. A projection method for unsteady Navier-Stokes equation with finite volume method and collocated grid. Hybrid Methods Heat Mass Transf. 1, 339–363. Sayers, A.T., 1987. Flow interference between three equispaced cylinders when subjected to a cross flow. J. Wind Eng. Ind. Aerodyn. 26 (1), 1–19. Sayers, A.T., 1990. Vortex shedding from groups of three and four equispaced cylinders situated in a cross flow. J. Wind Eng. Ind. Aerodyn. 34 (2), 213–221. Sumner, D., Price, S.J., Paidoussis, M.P., 2000. Flow-pattern identification for two staggered circular cylinders in cross-flow. J. Fluid Mech. 411, 263–303. Sumner, D., 2010. Two circular cylinders in cross-flow: a review. J. Fluids Struct. 26, 849–899. Sumner, D., Richards, M.D., Akosile, O.O., 2005. Two staggered circular cylinders of equal diameter in cross-flow. J. Fluids Struct. 20 (2), 255–276. Sumner, D., Wong, S.S.T., Price, S.J., Païdoussis, M.P., 1999. Fluid behavior of side-byside circular cylinders in steady cross-flow. J. Fluids Struct. 13 (3), 309–338. Tatsuno, M., Amamoto, H., Ishi-I, K., 1998. Effects of interference among three equidistantly arranged cylinders in a uniform flow. Fluid Dyn. Res. 22 (5), 297–315. Thapa, J., Zhao, M., Cheng, L., Zhou, T., 2015. Three-dimensional simulations of flow past two circular cylinders in side-by-side arrangements at right and oblique attacks. J. Fluids Struct. 55, 64–83.

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