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Three-dimensional numerical simulation of the flow around two cylinders at supercritical

Reynolds number

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2013 Fluid Dyn. Res. 45 055504

(http://iopscience.iop.org/1873-7005/45/5/055504)

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Fluid Dyn. Res. 45 (2013) 055504 (22pp) doi:10.1088/0169-5983/45/5/055504

Three-dimensional numerical simulation of the flowaround two cylinders at supercritical Reynoldsnumber

H X Hu, C B Liu, H Z Hu and Y G Zheng

State Key Laboratory for Corrosion and Protection, Institute of Metal Research, ChineseAcademy of Sciences, 62 Wencui Road, Shenyang 110016, People’s Republic of China

E-mail: hxhu@imr.ac.cn

Received 23 January 2013, in final form 31 July 2013Published 20 August 2013Online at stacks.iop.org/FDR/45/055504

Communicated by H J Sung

AbstractNumerical study of the flow past two tandem cylinders is carried out atP/D = 1.5 and 2.5 for Re = 2.8 × 105–7.0 × 105. The shear-stress transportk–ω turbulence model is selected to capture the flow characteristicsaround the cylinders. This paper focuses on the characteristics of thefluid field, hydrodynamic forces and vortex-shedding frequencies at twocylinder configurations for different Reynolds numbers (Re). Qualitative andquantitative comparisons with the published data are performed to evaluate thecurrent results and reasonable agreement is obtained. The results show thatvortex shedding occurs behind both the upstream and downstream cylinders atP/D = 2.5 for the entire region of testing Re, which is significantly differentfrom the most critical gap spacing (above 3D) at low Re. The drag directionchanges from negative at P/D = 1.5 to positive at P/D = 2.5. And thefluctuations found in the lift for the downstream cylinder are more drastic thanthat for the upstream cylinder, which indicates that the downstream cylindermay behave in large vibration. The Strouhal number (St) at P/D = 1.5 isrelatively low compared to that at P/D = 2.5 due to the strong interactionsbetween two cylinders with small gap spacing.

(Some figures may appear in color only in the online journal)

Nomenclature

D diameter of the cylinder (m)P distance between the centers of two cylinders (m)ρ density of the fluid (kg m−3)

© 2013 The Japan Society of Fluid Mechanics and IOP Publishing Ltd Printed in the UK0169-5983/13/055504+22$33.00 1

Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

L length of the cylinder along the spanwise direction (m)u streamwise velocity (m s−1)µ kinetic viscosity (kg ms−1)k turbulence kinetic energyω specific dissipation rateCd drag coefficientCl lift coefficientσ Prandtl numberSt Strouhal numberRe Reynolds number

Subscripts

1 upstream cylinder2 downstream cylinder

1. Introduction

Flow-induced vibration (FIV) is a common problem resulting in the failures of the heart-exchange tubes in the formation of fretting wear in a steam generator (SG) of nuclear powerplants. This type of failure is difficult to find and impossible to repair due to the complexstructure and special operation environment. FIV is highly dependent on the flow field aroundthe tube systems where a basic and classic issue is the flow past the tandem cylinders atsupercritical Reynolds number (Re). This paper focuses on the hydrodynamic studies on thesystem of flow past circular cylinders, although the coupling effect between the structure andthe flow is an important aspect in the studies of FIV.

The study of the flow past cylinders is old fashioned, but its complexity andpotential industrial application are still attracting researchers (Zhao et al 2012). Numerousinvestigations have been carried out on the flow around double and more cylindersconfigurations experimentally (Sumner et al 2000, 2005, Park 2003, Lee et al 2004, Okajimaet al 2007) and numerically (Mittal et al 1997, Meneghini et al 2001, Deng et al2006, Papaioannou et al 2006, Ding et al 2007, Kitagawa and Ohta 2008, Palau-Salvadoret al 2008, Harichandan and Roy 2010) and some achievements have been obtained. Threeflow regimes were classified in terms of the gap between the cylinders by Zdravkovich (1987).At small gap spacing (6 1.2–1.8D (D, diameter of the cylinder)), the shear layers separatingfrom the upstream cylinder roll up behind the downstream cylinder and form a vortex street. Atmoderate spacing (<3.4–3.8D), the upstream shear layers reattach the downstream cylinder.Vortex shedding occurs behind the downstream cylinder due to the flow separation from itssurface. At large spacing (>4D), periodical vortex shedding exists behind both the upstreamand downstream cylinders.

In addition to the studies of the effects of gap spacing on flow regime, a series of workspaid attention to the flow natures at different Re, such as those at Re < 500 (Carmo andMeneghini 2006, Ding et al 2007), Re < 1000 (Papaioannou et al 2006, Palau-Salvadoret al 2008) and Re < 104 (Kitagawa and Ohta 2008). As Re increases to the supercritical(3 × 105 < Re < 3.6 × 106), separating angles are delayed to the back of the cylinders andthe thickness of the laminar layer near the cylinder surface is greatly decreased. It is reportedthat the three-dimensionality of the flow become obvious when Re exceeds 190 (Deng et al2006, Papaioannou et al 2006). Hydrodynamic forces and turbulence transmission are more

2

Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

violent at high Re than those at low Re (Carmo and Meneghini 2006, Papaioannou et al 2006).As a result, the studies of complex flow structures have more potential applications at high Re.However, the investigations are extremely rare at the corresponding Re, apart from the studiesof Kassera and Strohmeier (1997) and Catalano et al (2003). Furthermore, most practicalindustrial environments are extremely complex with high turbulence intensity, which furtherpromotes studies on the flow past cylinders at high Re.

It is essential to figure out the flow properties at Re from 2.8 × 105 to 7.0 × 105

corresponding to the practical conditions in SG of a nuclear power plant. However, mostprevious work has been carried out below this Re region, as aforementioned. How doesthe flow behave comparing with that at low Re? What are the differences between thehydrodynamic forces at high and low Re? How does the flow regime vary for the samecylinder configurations at high Re? With these questions, a typical two-cylinders system intandem configurations with P/D = 1.5 and 2.5 was selected to clarify the flow profiles inSG of nuclear power plants in this paper, where P is the distance between the centers of thecylinders and D is the diameter of the cylinder. Three-dimensionality of the flow and thehydrodynamic forces were especially noticed and represented.

2. Methodology

2.1. Numerical scheme

The selection of turbulence model was greatly deliberated among k–ε, large eddy simulation(LES) and shear-stress transport (SST) k–ω models. Compared with the k–ε model, thek–ω model needs less computational time due to the absence of the wall functions and hassatisfactory numerical stability. On the other hand, the LES model can also precisely simulatethe cross flow field around cylinders at high Re (Kassera and Strohmeier 1997). However, thismethod takes an immense amount of computational time to adjust the Smagorinsky constantCs, an empirical parameter which differs from one flow case to another. Although the k–ω

model based on the Boussinesq approximation lacks accuracy for flows over curved surfaces,it is sufficient enough to predict the flow field in the wake of tube arrays, hydrodynamicforces and Strouhal number (St). In fact, it is a compromise between the computational timeand accuracy.

Pressure-based solver was selected to solve the fluid equations. The pressure–velocitycoupling equations were solved by SIMPLEC method. The gradient discretization wasperformed by the least squares method to ensure the second-order interpolation on non-orthogonal grids. Standard interpolation scheme and secondary order upwind scheme wereutilized to discretize the pressure and momentum equations, respectively. The time step wasset to 0.0006 s. The simulation was carried out by commercial ANSYS-FLUENT softwarepackages of an authorized edition.

Transport equations for the SST k–ω model are shown as follows:

∂

∂t(ρk) +

∂

∂xi(ρkui ) =

∂

∂x j

(0k

∂k

∂x j

)+ G ′

k − Yk, (1)

∂

∂t(ρω) +

∂

∂xi(ρωui ) =

∂

∂x j

(0ω

∂ω

∂x j

)+ Gω − Yω + Dω, (2)

0k = µ + ρkωσk

and 0ω = µ + ρkωσω

represent the effective diffusivity of k and ω, where

0k = µ +ρk

ωσk, (3)

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Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Table 1. Model constants.

σk,1 σk,2 σω,1 σω,2

1.176 1.0 2.0 1.168

0ω = µ +ρk

ωσω

, (4)

σk =1

F1/σk,1 + (1 − F1) /σk,2, (5)

σω =1

F1/σω,1 + (1 − F1) /σω,2, (6)

G ′

k = min(Gk, 10ρkω) represents the generation of turbulence kinetic energy due tomean velocity gradients, where Gk is given by Gk = −ρu′

i u′

j∂uj

∂xi.Gω = Gk

ωk represents the

production of turbulence kinetic energy. Yk = ρβ∗kω and Yω = ρβω2 represent the dissipationof k and ω due to turbulence, respectively.

Dω represents the cross-diffusion term, which is defined as

Dω = 2 (1 − F1) ρ1

ωσω,2

∂k

∂x j

∂ω

∂x j, (7)

where

F1 = tanh(84

1

), (8)

81 = min

[max

( √k

0.09ωy,

500µ

ρy2ω

),

4ρk

σω,2 D+ω y2

], (9)

D+ω = max

[2ρ

1

σω,2

1

ω

∂k

∂x j

∂ω

∂x j, 10−10

], (10)

y is the distance to the next surface and D+ω is the positive portion of the cross-diffusion term.

σk,1, σk,2, σω,1 and σω,2 are empirical constants for turbulence models, which are illustrated intable 1.

The streamwise force coefficient (Cd) and transverse force coefficient (Cl) were definedby two equations below, respectively:

Cd =Fx

0.5ρu2 DL, (11)

Cl =Fy

0.5ρu2 DL, (12)

Fx and Fy represents the forces acting on the cylinders along the streamwise andtransverse directions, respectively.

The details of parameters are listed in nomenclature.

2.2. Description of simulation conditions

Parameters used in the simulation were set according to the practical conditions includingthe fluid physical properties and velocities. The operation pressure was 8 MPa and the

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Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Figure 1. Schematic and the coordinate of the calculation domain.

corresponding density and dynamic viscosity of the liquid flow were 832.2 kg m−3 and1.17 × 10−4 kg ms−1, respectively. The diameter of the cylinder was 0.02 m, which wasthe same as that of the heat-exchanger tubes in SG of a nuclear power plant. The inletand outlet boundary conditions were velocity inlet and pressure outlet, respectively. Fourvelocities namely 2, 3, 4 and 5 m s−1 were set as the inlet velocities, which were withinthe practical velocity region in SG. The corresponding Re were approximately 2.8 × 105,4.2 × 105, 5.5 × 105 and 7.0 × 105, respectively. Two cylinders in tandem arrangement withP/D = 1.5 and 2.5 were selected for the simulation, which are the basic arranging units oftube bundles in SG. Geometry of the calculation domain with the arrangement of cylinderis shown in figure 1. Two equal-sized circular cylinders reside in the rectangular calculationdomain whose upstream boundary is located at a distance of 10D from the center of theupstream cylinder. The entire length of the computational domain is 30D in the streamwisedirection. The lateral boundaries located at 10D distance each from the upstream cylinder andtwo side faces together constituted the flow passage. The lengths of the cylinders are identicalto that of Breuer (1998, 2000). The standard no-slip boundary condition is specified on thecylinder wall.

Mesh generation was performed by Gambit, which is a commercial software packagedesigned to build geometry models and generate meshes for CFD. Take the case of P/D = 1.5for instance; the calculation domain is divided into 16 parts as shown in figure 2(a). Thepurpose is to precisely capture the flow near cylinders and properly reduce the amount ofcalculation. The mesh details near cylinders and side walls are shown in figure 2(b). Boundarylayer meshes are utilized in the near pipe wall region (parts of 1–8). It is 0.05 mm from theinitial node to the cylinder wall surface. Relatively coarse meshes distribute in the places farfrom the cylinders. However, these meshes are intensive adjacent to cylinders and graduallysparse far away from the cylinders. The first length of the grid is 1.2 mm and increases by aconstant ratio of 1.01 in X and Y directions (parts of 9–16). The meshes are uniform with alength of 4 mm in Z direction.

3. Convergence test and validation

A three-dimensional simulation for the flow past an isolated cylinder at Re ≈ 1.4 × 105 wascarried out to validate the numerical method and the grid system. The Re above was selected

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Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Figure 2. Grid system in the vicinity of the cylinders and the close-up view of the grid on thecylinder wall.

to match the data in the similar study conducted by Cantwell and Coles (1983) at the same Re.The following simulations of flow past tandem cylinders adopted the same numerical methodand grid scheme. Grid dependency check was carried out by comparing the results from threecell densities at Re = 2.48 × 105, 7.88 × 105 and 1.2 × 106 with previous experimental data(Cantwell and Coles 1983).

Figure 3 shows comparison of the mean streamwise velocity (X-velocity) distributionalong the line Y = 0 in XY plane with previous experimental results. The origin (0, 0, 0) islocated at the center of the upstream cylinder. One grid check was also performed in termsof the streamwise velocity distribution plotted as the different X/D locations. In the first 2Ddistance for case 1 and case 2, the mean streamwise velocity of our simulation agreed verywell with the experimental data of Cantwell and Coles (1983). In the far field, there is a slightover-prediction of the present X-velocity from those given by Cantwell and Coles (1983).

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Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Figure 3. Time-averaged velocity in the streamwise direction along the line y = 0, z = π D fordifferent grid systems.

Table 2. Overview of the simulations of different grid systems.

Pattern Case 1 Case 2 Case 3 Cantwell and Coles (1983)

Number of cells 248 000 (31% case 2) 788 000 1 258 000 (160% case 2) –St 0.252 0.174 0.256 0.179

Moreover this deviation decreased with increasing distance from the cylinder. In contrast, thepredicted result for case 3 apparently deviated from the experimental data almost in the entiretested distance. Therefore, the grid regimes of case 1 and 2 are better than that of case 3 inthe view of velocity distribution. Another grid check was carried out by comparing the St andthe results are presented in table 2. Out of the three cases, St from case 2 is closer to thosereported by Cantwell and Coles (1983).

Preliminary tests indicate that there is no fixed relationship between the grid density andSt, which are not presented here. The variations of St with the grid cells number may bepossibly related to the differences of mesh densities in the near-wall region and the regionfar away. Further studies are still needed to figure out the reasons leading to the variationof St, which is not the emphasis of the paper. Through the collaborative examinations of thevelocity and St distributions, the grid generating strategy of case 2 is used for the followingsimulations.

Before evaluating our present grid system and flow models, a relationship must be noticedin comparison between the simulation and experiments. In simulation, the predicted resultsare highly dependent on some parameters, such as flow models, discretization methods,the size of calculation domain and the boundary conditions. In the case of experiments,the data have strong dependence on the experiment set-up. Moreover, the problem of flowpast cylinders at high Re is extremely complex due to aspects including the arrangement ofcylinders, blockage ratio, Re and roughness of the cylinder surfaces. All these reasons lead tothe scattered data collection and its disagreement of St with those reported by Cantwell andColes (1983), Zdravkovich (1997) and Fey et al (1998). Despite a slight deviation from theexperiments, our predicted simulation results are found to be reasonably good.

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Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

4. Results

Few valuable data can be used to evaluate the simulation results at the given Re (= 2.8 × 105,4.2 × 105, 5.5 × 105, 7.0 × 105), correspondingly. However, some experiment and simulationdata at subcritical and supercritical Re are utilized for the rough comparison. One purpose isto evaluate our simulation results, and the other is to present the flow discrepancy betweenat low and high Re. All the results are collected from the stable stage of the calculation.Typical characteristics of the flow past the cylinders are revealed, such as the vortex evolution,hydrodynamic forces and St.

4.1. Flow field

4.1.1. Velocity distribution. The case for Re = 2.8 × 105 is selected as a representation toinvestigate the three-dimensionality of the flow at high Re. Five section planes, namely Z = 0,15.7, 31.4, 47.1 and 62.8 are used to reveal the velocity variations along the streamwise andspanwise directions. The velocity distributions on the section planes are shown in figure 4 oninstantaneous dimensionless time. At P/D = 1.5 (figure 4(a)), three-dimensionality mainlyoriginates from the site 3D from the center of the downstream cylinder, which is consistentwith that reported by Deng et al (2006). Velocity distributions on the section planes differfrom each other suggesting an obvious three-dimensionality in the spanwise direction at highRe. Moreover, the velocity contours before the upstream cylinder are almost the same forall the section planes as expected, which indicates that existence of the downstream cylinderstrongly suppresses the three-dimensional instability.

Compared with the case for P/D = 1.5, apparent periodicity is observed in the velocitydistribution at P/D = 2.5 (figure 4(b)). High-velocity vortices shed from the cylinders andgradually dissipate in the far field downstream in the streamwise direction. The velocitycontours are dominated by the two-dimensionality before the upstream cylinder deduced fromthe same velocity distribution there, which is consistent with that for P/D = 1.5. On the otherhand, slight difference occurs in the gap between the cylinders. The region with low velocityjust behind the downstream cylinder is much larger at the middle section plane of Z = 31.4than that at the other planes, which is a signature of the three dimensionality. Moreover, thevelocity distributes symmetrically with the section plane of Z = 31.4, i.e. the velocity at thesection plane of Z = 0 and 1.57 are the same as that of Z = 62.8 and 47.1, respectively. Itreveals the velocity periodicity in the spanwise direction as well as the three-dimensionality.

Figures 5(a) and (b) show three-dimensional views of the velocity iso-surfaces for theflow past tandem cylinders at P/D = 1.5 and = 2.5 for Re = 2.8 × 105, respectively. It can beobserved that greatly distorted iso-surfaces indicate high three-dimensional flow nature in thewake of the downstream cylinder (figure 5(a)), which is consistent with the velocity profiles(figure 4(a)). The flow nature of two-dimensionality is well presented by the smooth andregular curve surfaces before the upstream cylinder. There is no strong energy transportationbetween the velocity iso-surfaces without the disturbance of the upstream cylinder, whichis similar to the case of P/D = 2.5. In contrast, considerably drastic turbulence occursat P/D = 2.5 (figure 5(b)). Vortex shedding phenomenon is composed of the multilayerstructures with high velocity gradient behind the cylinders. The fluctuation observed in theiso-surfaces in the spanwise direction clearly reveals the three-dimensional structures of thevortex. The phenomenon is very similar to that reported by Carmo at P/D = 1.5 for Re = 200(Carmo and Meneghini 2006). Moreover, the iso-surfaces are twisted more significantly in thefar field than that in the near wake of the downstream cylinder. Therefore, it can be deducedthat the three-dimensionality becomes relatively drastic on evolution of the vortices.

8

Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Figure 4. Velocity contours at different planes in the Z direction at (a) P/D = 1.5 and(b) P/D = 2.5 for Re = 2.8 × 105.

Figures 5(c) and (d) present velocity traces on the cylinder walls at P/D = 1.5 and2.5, respectively. At P/D = 1.5 (figure 5(c)), the surface streamlines joint, tangle, anddisperse irregularly near the gap sides on the walls, which reveals the chaotic flow attachmentand hydrodynamic forces on the cylinders. Moreover, some streamlines are nearly parallelwith the spanwise direction, which is more obvious than that in the corresponding regionfor the case of P/D = 2.5 (figure 5(d)). It is mainly caused by the great interaction betweenthe cylinders at the configuration with P/D = 1.5. In contrast, most surface streamlines crossthe cylinders in parallel at the regime of P/D = 2.5 (figure 5(d)). Only slight variations locateon the rear cylinders due to the separation of the shear layers. These uneven distributions againprove the effect of the vortices on the cylinders and the three-dimensionality of the flow athigh Re.

4.1.2. Vortices distribution. Figure 6 displays the evolution of the vorticity contours in acycle at P/D = 1.5 and 2.5 for Re = 2.8 × 105. Five instantaneous times with equal timeintervals, namely A, B, C, D and E, are selected to reveal the variation of the vortex with

9

Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Figure 5. Velocity distributions on the iso-surfaces and the cylinders walls at (a) and (c) P/D =

1.5 and (b) and (d) P/D = 2.5 for Re = 2.8 × 105.

time. These points in figures 6(a) and (c) assist in understanding the flow distribution withtime rather than indicating their periodicity. At Y = 0 plane for P/D = 1.5 (figure 6(a)), theflow separates simultaneously at both the upper and lower sides of the upstream cylinder,reattaches and rolls over the downstream cylinder without the formation of vortex in thegap spacing. These phenomena have been clarified as the bistable in the investigation ofZdravkovich and Pridden (1977), which also existed at low Re (= 2.2 × 104) (Kitagawa andOhta 2008). It indicates the subcritical Re does not change the flow regime at the settledcylinder configuration of P/D = 1.5. Random vortex shedding is observed in the wake ofdownstream cylinder. However, it is not of periodicity and the vortex shape is elongatedirregularly. As the gap increases to P/D = 2.5 (figure 6(b)), distinct vortices shed from boththe upstream and downstream cylinders periodically and regularly. The wake flow of thedownstream cylinder consists of two rows of alternated Karman vortex streets. One shedsfrom the upper sider, and the other is from the lower side of the cylinder. The vorticesin the gap between two cylinders reattach the downstream cylinder before shedding fromthe upstream cylinder. Double vortex shedding phenomenon is well consistent with that atthe same cylinder configurations for Re = 1000 reported by Jester and Kallinderis (2003).The vorticity contours at corresponding time in the spanwise direction are displayed infigures 6(c) and (d), respectively. In figure 6(c), one can see that the vortices vary randomlywith time in the gap spacing. The highly anisotropic turbulence is mainly due to the interaction

10

Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Figure 6. Vorticity contours in a cycle time around the tandem cylinders in Z = 0 and Y = 0 planesat (a) and (c) P/D = 1.5 and (b) and (d) P/D = 2.5 for Re = 2.8 × 105.

between the closer cylinders at high Re. There still a rule can be obtained, although thevorticity distributions are chaotic. It is that the vorticity is high in the near wake of thedownstream cylinder and dissipates with the distance extension to the far field. As the gapincreases to 2.5D, evident periodicity can be observed in both the streamwise and spanwisedirections (figure 6(d)). The images clearly present the vortex evolution processes includingtheir alternated generating, shedding and dissipating in the wake of the downstream cylinder.Moreover, the uneven distribution along the cylinder also reveals the three-dimensionality ofthe flow in the view of vorticity distribution at high Re.

Figures 7(a) and (b) show the three-dimensional views of the vorticity in the formationof iso-surfaces for the flow past tandem cylinders at P/D = 1.5 and = 2.5, respectively. Thethree dimensionality of the vorticity field is better represented in these profiles. Comparingfigure 7(a) with figure 7(b), it can be observed that the cylinders cause turbulence moredrastically at P/D = 2.5 than at P/D = 1.5 and the vortices transmit relatively far. However,they have in common that the two-dimensional flow structure dominates the flow before theupstream cylinder both for P/D = 1.5 and 2.5, which is very similar to that at P/D = 1.5 forRe = 200 observed by Carmo and Meneghini (2006).

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Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Figure 7. Distributions of the vorticity iso-surfaces at (a) P/D = 1.5 and (b) P/D = 2.5 forRe = 2.8 × 105.

4.2. Hydrodynamic forces

4.2.1. Drag and lift. Figures 8(a) and (b) plot the time history of the drag coefficients for alltested Re at P/D = 1.5 and = 2.5, respectively. The numbers 1 and 2 after Cd represent theupstream and downstream cylinders, respectively. The data were collected in the stable regionof the drag history. In figure 8(a), both the upstream and downstream cylinders experiencenon-periodic drag at P/D = 1.5 for all the cases. It is much higher acting on the upstreamcylinder than that on the downstream cylinder, but its fluctuation is relative small compared tothat for the downstream cylinder. This is due to the shielding effect of the upstream cylinderat smaller gap spacing. Moreover, the downstream cylinder suffers a negative force becauseit immerses in the negative pressure region caused by the shear layer from upstream cylinder.It is consistent with that observed at Re = 200 by Ding et al (2007). As a result, it can bededuced that the drag direction for the downstream cylinder is determined by the gap betweenthe cylinders, not by Re at P/D = 1.5. As the gap increases to 2.5D (figure 8(b)), the dragfluctuates periodically for both the upstream and downstream cylinders and its amplitude ismuch higher than that at P/D = 1.5. The drag for the upstream cylinder is larger than thatfor the downstream cylinder, which is the same as the case for P/D = 1.5. It is clear that thedownstream cylinder is still under the unsteady wake of the upstream cylinder. Moreover,when the drag is highest for the upstream cylinder, it falls down to the bottom for thedownstream cylinder indicating an anti-phase difference. This trend does not change withincreasing Re from 2.8 × 105 to 7.0 × 105.

The lift variations for the tandem cylinders with time at different Re for P/D = 1.5and 2.5 are shown in figure 9. The numbers 1 and 2 after Cl represent the upstream anddownstream cylinders, respectively. Comparing with the drag (figure 8), the amplitude of thelift is very high. For the upstream cylinder at P/D = 2.5, the amplitude for the drag coefficientis about 0.05, which is only 1/14 of that for the lift coefficient. Large lift often causes largevibration, which is why vibrations are more drastic in the transverse direction than those inthe streamwise direction.

There are significant differences between the lift coefficients at P/D = 1.5 and 2.5. Thefluctuations found in the lift are irregular for the entire Re at P/D = 1.5. In figure 9(a),it is temperate suggesting that there is no large vortex shedding within the period fromUt/D = 400 to 430. It becomes very drastic from Ut/D = 430 to 460 and then moderatesfollowing the second drastic fluctuations from Ut/D = 470 to 490. This lift history indicatesthat large vortex shedding exists casually resulting in the casual increase in the amplitude ofthe lift, which is shared by the cases for the entire Re at P/D = 1.5. When the vortex sheddingis stable and continuous, the fluctuations in the lift will be regular and periodic. As a result,

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Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Figure 8. Time history of the drag coefficient for the upstream and downstream cylinders at (a),(c), (e) and (g) P/D = 1.5 and (b), (d), (f) and (h) P/D = 2.5 for Re = 2.8 × 105, 4.2 × 105,5.5 × 105 and 7.0 × 105.

they are stable and periodic at P/D = 2.5 as expected (figure 9(b)). Moreover, the fluctuationsfound in the lift for the downstream cylinder are larger than that for the upstream cylinder atboth P/D = 1.5 and = 2.5. While these fluctuations observed in the upstream cylinder arecaused by movement of the shear layer, they are induced by the alternate impingement ofvortices from the upstream cylinder, in the case of the downstream cylinder. As a result, itcan be deduced that the irregular fluctuations found in the lift for the downstream cylinderresult from the unsteady wake flow due to the upstream cylinder at P/D = 1.5. Regularfluctuations observed in lift correspond to the periodic vortex shedding from both the cylindersat P/D = 2.5. In addition, the oscillation for the downstream cylinder is reasonably largerthan that for the upstream cylinder, which is consistent with that reported by Meneghini et al(2001) and Prasanth and Mittal (2009).

Meneghini et al (2001) numerically obtained the drag and lift time histories for the flowpast tandem cylinders with P/D = 1.5 at low Re. They found that the drag acting on the twocylinders was completely constant and the lift fluctuated periodically. As Re was increasedto 200, they found large fluctuations in the fluid forces. In contrast, the fluctuations observedin the fluid forces are relative large and irregular for the same arrangement of the cylinders

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Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Figure 9. Time history of the lift coefficient for the upstream and downstream cylinders at (a),(c), (e) and (g) P/D = 1.5 and (b), (d), (f) and (h) P/D = 2.5 for Re = 2.8 × 105, 4.2 × 105,5.5 × 105 and 7.0 × 105.

at high Re in current study. The discrepancy indicates that the fluid instability increases withincrease in Re.

Figure 10 presents the variations of averaged and maximum peak of the drag and liftcoefficients with Re for the upstream and downstream cylinders. In figure 10(a), the averageddrags acting on the upstream cylinder are higher than those on the downstream cylinder for theentire testing Re, and they almost do not change with Re at both P/D = 1.5 and 2.5. For theupstream cylinder, it is higher at P/D = 2.5 than that at P/D = 1.5, which is also applicableto the downstream cylinder. The maximum peak drag coefficients behave in the same way asthat of drag with increasing Re (figure 10(c)).

The variations of the averaged lift are different from that for the drag. At P/D = 1.5in figure 10(b), the averaged lift coefficient for the upstream cylinder sharply increases from−0.002 at Re = 2.8 × 105 to 0.002 at Re = 7.0 × 105. While, it stays above zero at the firsttwo Re at P/D = 2.5, which indicates that the fluctuations observed in the lift prefer to vibratethe cylinder in the positive y direction. And then drop down to zero at the last two Re. Theentire lift variation with Re for the upstream cylinder is the same as that for the downstreamcylinder, which suggests that the vortex shedding in the gap is synchronized with that in thewake of the downstream cylinder.

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Fluid Dyn. Res. 45 (2013) 055504 H X Hu et al

Figure 10. Dependence of (a) the averaged drag, (b) averaged lift, (c) maximum peak drag and(d) maximum peak lift coefficients on Re at P/D = 1.5 and 2.5.

Figure 10(d) shows the relationship between the maximum peak value of the lift andRe for cylinders at P/D = 1.5 and = 2.5. It can be noted that the amplitude of lift nearlyremains in a stable value with increasing Re, but behaves distinctly from each other. For Re =

2.8 × 105, it is about 0.9 at P/D = 2.5, which is 5.6 times larger than that (approximately0.16) at P/D = 1.5. This indicates that the vortex shedding in the gap at P/D = 2.5 (figure 6)significantly stimulates the oscillation of the upstream cylinder as aforementioned. For thesame reason to the downstream cylinder, the amplitude is about 1.65 at P/D = 2.5, which is3.7 times higher than that (nearly 0.45) at P/D = 1.5. The great distinction in the peak liftsuggests that the cylinders are vibrated by the dramatic fluctuated lift.

4.2.2. Skin friction. The mean skin friction along circumference on the upstream cylinderwall is used to compare with the published data as shown in figure 11. The circumference is inthe middle of the cylinder wall with the coordinate of Z = 31.4 mm. Like the distributions ofCp (figure 12) and streamwise velocity (figure 13), the skin friction coefficient at P/D = 2.5also behaves very similarly to that at P/D = 1.5. In addition, remarkable discrepanciesare observed between the present results and the reference data in the front half of thecylinder. The peak value of the present skin friction coefficient is lower than that reportedby Catalano et al (2003) and Achenbach (1968), but higher than that observed by Travin et al(1999) probably due to the differences in Re, experiment and simulation sets. In contrast,

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Figure 11. Comparison of the mean skin friction coefficient distributions for the upstream cylinderbetween current results and the published results (Achenbach 1968, Travin 1999, Catalano 2003).

Figure 12. Mean pressure coefficient distribution on the surface of the upstream cylinder(Warschauer and Leene 1971, Cantwell and Coles 1983, Norberg 1987, Kassera and Strohmeier1997, Breuer 1998).

our simulation results at Re = 2.8 × 105 reasonably fall between the data at Re = 1.4 × 105

and 3.6 × 106. This indicates that the present model is slightly superior in capturing thelaminar layer on the surface of the cylinder, although there is deviation from the referencedata. Moreover, the simulated data are roughly consistent with that on the back half of thecylinder observed by Catalano et al (2003). It is clear that an acceptable capturing of theflow separation is obtained in the wake of the upstream cylinder. Very good agreement withTravin et al (1999) is achieved in predicting the flow separation in the lower half of thecylinder.

4.3. Separation angles

The flow past circular cylinders will experience a series of processes including impact, rollingand separating. The separating angle is related to Re, which can be obtained through some

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Figure 13. Mean streamwise velocity distributions on the wall surfaces of (a) the upstream and(b) downstream cylinders at P/D = 1.5 and 2.5 for Re = 2.8 × 105.

parameters, such as the mean pressure coefficient and the surface velocity distribution on thecylinder walls.

Figure 12 compares the current simulation result with the published data in Cp

distributions on a circumference of the upstream cylinder. The correlation data were extractedfrom the references to evaluate the simulation results. The circumference locates in thecoordinate of Z = 31.4 mm on the cylinder wall. The case of Re = 2.8 × 105 is selected tomatch the reference data due to their similar turbulence intensity. Three types of distributionscan be observed in the view of Re in the plot. The first one is at lower Re (3 × 103–1.4 × 105),which includes those reported by Norberg (1987), Breuer (1998) and Kassera and Strohmeier(1997). The corresponding separation angle is around 80◦. The second one is at higher Re(6.7 × 106–1.20 × 107) including Warschauer and Leene (1971) and Kassera and Strohmeier(1997), where Cp is much lower than that at low Re within the angles from 60◦ to 120◦.The separation angle is significantly delayed to about 120◦. The last one is present resultat Re = 2.8 × 105 for P/D = 1.5 and 2.5, which locates in the middle of two types above.The Cp distributions are almost the same for the two cylinder configurations. Both thecorresponding separation angles are approximately 117◦.

Figures 13(a) and (b) show the distributions of the mean streamwise velocity on thecircumference of Z = 31.4 for the upstream and downstream cylinders, respectively. For theupstream cylinder (figure 13(a)), the mean streamwise velocity at P/D = 1.5 behaves nearlythe same as that at P/D = 2.5, like the pressure coefficient distributions (figure 12). It canbe deduced that the existence of the downstream cylinder has few effects on the velocitydistributions on the upstream cylinder. The separation angle where the flow direction reversesis approximately 120◦ for both the cylinder configurations, which slightly over-predicts itagainst that (117◦) obtained by Cp distribution (figure 12). For the downstream cylinder(figure 13(b)), the velocity at P/D = 1.5 distributes differently from that at P/D = 2.5resulting from the completely different flow distribution in the gap as shown as figures 4–7.Moreover, the site where the peak velocity occurs at P/D = 1.5 is slightly late compared tothat at P/D = 2.5. However, the separation angles are the same for them, which locate atabout 142◦. The separation angle for the downstream cylinder is much later than that for theupstream cylinder because the separation layer has become turbulent due to the disturbing ofthe upstream cylinder at subcritical Re.

There are some disputes in the definition of the separation angle. Some researchersdetermined the separation angle based on Cp. They deem that the separation angle locates

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at the point where Cp gets into a stable period. Cantwell and Coles (1983) found the inflectionposition at 77◦ at Re = 1.4 × 105 in their experiments. However, some studies proposedthat the separation angle locates at the point where the direction of the streamwise velocityreverses from negative to positive. Breuer (2000) reported that it is 94◦ at Re = 2.6 × 105 inhis research. Some data are evidently different from that observed by Breuer (2000), whichare about 78◦ at Re = 105 (Son and Hanratty 1969) and 72◦ at 1.5 × 105 (Achenbach 1968).Therefore, most separation angels are between 70◦ and 80◦ excepting the 94◦ reported byBreuer (2000). It is accepted that 3.0 × 105 is the critical Re, beyond which the separationangle is greatly delayed. However, the separation angles are different from each other, evenat the same Re region as aforementioned. The discrepancy is possibly due to the differencesin the simulation and experiment conditions. As a result, it is reasonable that the currentseparation angle is also delayed to about 120◦, although Re does not exceed the critical Re.It is mainly due to the high turbulence intensity of the flow in the gap resulting from thedisturbing from the downstream cylinder.

4.4. Strouhal number

St refers to non-dimensional frequency of vortex shedding, which is defined as St = fs D/U∞

(Rockwell 1998), where fs is the dominant frequency of the lift coefficient obtained byperforming the fast Fourier transform and U∞ is free stream velocity.

Figure 14 presents the variation of power spectra density with increasing Re at P/D = 1.5and 2.5. St for the upstream cylinder is the same as that for the downstream cylinder.Therefore, St in these images represents for that of both cylinders corresponding to eachcondition. It can be observed that there is only one largest peak value of the spectrum, whichcorresponds to an St dominating the flow main frequency of the vortex shedding. However,the power spectra distribution is neater at P/D = 2.5 than at P/D = 1.5. In figures 14(a),(c), (e) and (g), several small peaks cluster around the main peak indicating disorderedvortex shedding processes at P/D = 1.5. It is consistent with the random variation of thelift coefficient (figures 9(a), (c), (e) and (g). In contrast, there is only one distinct andsharp peak at the test Re (figure 9(b), (d) and (h)), except for the case of Re = 4.4 × 105 atP/D = 2.5 (figure 9(f)). No small peak appears near the main peak of the power spectra. Thisunique phenomenon suggests a constant vortex shedding frequency, which is correspondingto the periodic variation of the lift coefficient. In addition, two peaks occur at Re = 4.4 × 105.The one with higher magnitude corresponds to dominant frequency of the vortex shedding.The other one with lower magnitude is a signal of small change in the periodic nature of thevortex shedding (Deng et al 2006, Patil and Tiwari 2008).

Table 3 compares the present St with reference data at different Re and cylinderconfigurations. At P/D = 1.5, most of the St is about 0.12 except for a sudden decreaseto around 0.04 at Re = 4.4 × 105, which is 1/3 of that at other Re. The small value indicatesa slow vortex shedding at the certain Re. As the gap increases to 2.5D, St rises to about 0.2which is obviously higher than that at P/D = 1.5 for the entire ranges of Re. It is interestedthat there are two peaks for the case of Re = 4.4 × 105 due to the small change of the vortexshedding. One (0.142) is lower and the other one (0.281) is higher than the average St (0.2) forother Re. St is lower at small gap than that for the single cylinder (Shih et al 1993, Catalanoet al 2003). But it is almost equal to the value when the gap increases to 2.5D. It can bededuced that St is strongly affected by the interactions between the two cylinders with smallgap. This conclusion is in good agreement with that reported by Meneghini et al (2001). Theyfound that St at P/D = 1.5 is also relatively low compared to that for the single cylinderat Re = 200. Similar observations can also be seen in these references (Deng et al 2007,

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Figure 14. Strouhal number at (a), (c), (e) and (g) P/D = 1.5 and (b), (d), (f) and (h) P/D = 2.5for different Re.

Wu et al 1994), which again proves that the existence of the adjacent downstream cylinderhighly suppresses St for the upstream cylinder at both the high and low Re. On the other hand,Re plays a rather restricted role in St in the range from Re = 105 to 106.

St values are not identical to each other for different Re as shown as table 3. Carmoand Meneghini (2006) proposed that St increased from approximately 0.15 to 0.19 as Reincreased from 160 to 320 in their simulation at P/D = 1.5. Deng et al (2006) studied thedependence of St on the P/D ratio and observed St = 0.17 at Re = 220 through their two- andthree-dimensional calculations. Moreover, Okajima et al (2007) and Alam et al (2003) alsodiscovered that St is small at the tandem cylinders configurations with narrow gap, which areSt = 0.14 for Re = 2.0 × 104 and 0.135 for Re = 6.5 × 104, respectively. From the reviewsof the investigations above on St, it can be deduced that the downstream cylinder highlysuppresses the vortex shedding frequency at the cylinder configuration with P/D = 1.5, andRe has few effects on Re. As the gap increases to P/D = 2.5, St is approximately 0.2 for allthe ranges of Re.

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Table 3. Summary of St at different Re and cylinder configurations.

Re 2.8 × 105 3.5 × 105 4.4 × 105 7.0 × 105

P/D = 1.5 0.132 0.126 0.04 0.126P/D = 2.5 0.209 0.212 0.142 and 0.281 0.221Travin et al (1999) 0.21 (single cylinder, Re = 1.5 × 105)Shih et al (1993) 0.22 (single cylinder, Re = 1.0 × 106)Catalano et al (2003) 0.35 (single cylinder, Re = 1.0 × 106)Meneghini et al (2001) 0.167 (tandem cylinders, P/D = 1.5, Re = 200)Okajima et al (2007) 0.14 (tandem cylinders, P/D = 1.5, Re = 2.0 × 104)Alam et al (2003) 0.135 (tandem cylinders, P/D = 1.5, Re = 6.5 × 104)Jester and Kallinderis (2003) 0.22 (tandem cylinders, P/D = 2.5, Re = 1.0 × 103)

5. Discussion

The flow past cylinders is a classic fluid dynamic issue, which has attracted numerousinvestigations due to its wide potential industry applications and scientific complexity. Theflow regimes have been clarified at different Re and accepted by most people. At Re from150 to 300, the boundary layer stays in a transition state from laminar to turbulence. Atsubcritical Re from 300 to 3 × 105, the flow is laminar near the cylinder, and turbulent in theseparation layer. The supercritical Re ranges from 3 × 105 to 3.6 × 106 where the separationpoint moves to the back of the cylinder and the width of the laminar is significantly decreased.As Re increases to ultra-supercritical exceeding 3.6 × 106, the flow has been turbulence beforeseparating.

In addition to Re, the cylinder configuration is the key aspect affecting the flow regimes.Co-vortex shedding occurs at P/D = 2.5, not at P/D = 1.5, which indicates that there isa critical cylinders regime between P/D = 1.5 and = 2.5. However, Deng et al (2006)found a critical P/D ratio between 3.5 and 4 at Re = 220. Kitagawa and Ohta (2008)found the similar regime of P/D = 3.25 at Re = 2.2 × 104 where the vortex shedding occursbehind both the upstream and downstream cylinders. The critical P/D ratio is even smallerin the experimental investigations performed by Ding et al (2007). He observed a quasi-steady reattachment regime at P/D = 2.5 for Re = 100 and 200. Others are P/D = 3.0for Re = 4.9 × 103 (KiYa et al 1980, Wu et al 1994) and P/D = 3.5 for Re = 2.2 × 104

(Kitagawa and Ohta 2008). It can be deduced that the critical gap spacing is nonlinear withRe, although they are at the same subcritical Re. In the present simulation, the critical P/Dratio is between 1.5 and 2.5, which is much smaller than that reported in most investigations.However, it is consistent with that proposed by Mittal et al (1997) at the same cylinderconfiguration. It seems that other aspects also have an effect on the flow regime except forthe gap and Re, such as the experimental set and simulation conditions.

Apart from the flow regimes, the three-dimensionality at high Re differs from thatat low Re. Highly anisotropic distribution observed in the spanwise direction shows thethree-dimensional flow nature. For P/D = 1.5, complete turbulence distributes in the gap(figures 5 and 7). In contrast, obvious periodic vortex structure occurs in both the spanwiseand streamwise directions at P/D = 2.5 (figures 5 and 7). However, it cannot be observedat P/D < 3 for Re = 200 according to the investigation carried out by Deng et al (2006).Therefore, it can be deduced that high Re further aggravates the turbulence intensity afterdisturbance by the cylinder. It causes and strengthens the turbulence in advance at thesame cylinder regimes. As a result, three-dimensional simulations are essential in preciselycapturing the flow structures and the hydrodynamic forces.

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6. Conclusions

The flow past the cylinders in tandem configurations with P/D = 1.5 and = 2.5 fromRe = 2.8 × 105 to 7.0 × 105 has been successfully simulated. Typical properties includingflow field, hydrodynamic forces and vortex shedding phenomena are roughly evaluated by theprevious results, although there is a lack of corresponding data for comparison. The presentwork is important for understanding the flow details in the SG of nuclear power plants. It ishelpful in investigating the FIV of the tube bundles.

(1) In the range of Re from 2.8 × 105 to 7.0 × 105, periodic vortex shedding exists behind thecylinders at P/D = 2.5, but not at P/D = 1.5. The critical gap spacing is smaller thanthat reported in most published data. However, it is well in accordance with the simulationresults for Re = 1000 at the same cylinder configuration with P/D = 2.5.

(2) Three-dimensional characteristic is weaker at P/D = 1.5 than at P/D = 2.5 due to thesuppression of the downstream cylinder at the close gap space. Periodic vortex evolutionoccurs in both the streamwise and spanwise directions at P/D = 2.5 for subcritical Re.

(3) The downstream cylinder suffers a negative drag due to locating in the low pressure regioncaused by the upstream cylinder at P/D = 1.5. As the gap increases to 2.5D, the dragbecomes positive and periodic vortex shedding results in the highly fluctuated lift.

(4) St is relatively low at P/D = 1.5 (approximately 0.12) compared to that at P/D = 2.5(approximate 0.2) due to the strong interactions between the two cylinders at the closegap. The small peaks in the power spectrum of St indicate chaotic flow shedding atP/D = 1.5. Re = 4.4 × 105 is a critical value where two dominant periodic vortexshedding frequency results in the double peaks in St at P/D = 2.5.

Acknowledgment

The authors acknowledge the financial support of the Special Funds for the Major State BasicResearch Projects (2011CB610504).

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