Classifications of Flow Pattern on Three Circular Cylinders in Equilateral-triangular Arrangement

Journal of Wind Engineering

and Industrial Aerodynamics 89 (2001) 553–568

Classifications of flow pattern on three circularcylinders in equilateral-triangular arrangements

Zhifu Gu*, Tianfeng Sun

State Key Laboratory for Turbulence Research, Department of Mechanics and Engineering Science,

Peking University, Beijing 100871, People’s Republic of China

Received 18 April 2000; received in revised form 22 October 2000; accepted 3 November 2000

Abstract

Classifications of flow pattern on three parallel circular cylinders in equilateral-triangular

arrangements in cross uniform flow were investigated by wind tunnel tests. The spacing ratioswere in the range of 1:74N=d45:0, where N is the distance between the centers of adjacentcylinders, d the diameter of the cylinders. Both the experiments of pressure distributions onthree circular cylinders at Reynolds number 5.5� 104 and flow visualizations at Reynolds

number 1.4� 104 were carried out. It shows that the angles of incident flow strongly influencethe flow patterns, and therefore the pressure distributions on any one of the cylinders. Due tothe interference of the shear layers and/or wakes, the most complex features of pressure

distributions can be found on the cylinder located downstream. As for the different levels ofinterference regarding the influence of spacing ratio, mainly four different affected regions, i.e.,effects of small, transition, medium, and larger spacing can be identified. Within the small

spacing ratios, i.e., 1:74N=d42:2, three basic types of interference can be classified. The threebasic types are interference of proximity, shear layer reattachment and wake. Furthermore,seven steps of flow pattern can be subdivided as the angles of incident flow vary from 08 to 608.# 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Circular cylinders; Pressure distribution; Flow visualization; Wake structure; Flow pattern

1. Introduction

In many cases of engineering practice, bodies often appear in the form of groups,e.g. groups of buildings, chimneys, stacks, chemical reaction-towers, offshore

*Corresponding author. Tel.: +86-10-6275-6079; fax: +86-10-6275-6079.

E-mail address: [email protected] (Z.F. Gu).

0167-6105/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 1 6 7 - 6 1 0 5 ( 0 0 ) 0 0 0 9 1 - X

platforms etc. Due to the mutual interference, the aerodynamic characteristics, suchas pressure distributions and vortex-shedding patterns on each member in the groupare completely different from those of an isolated one.

A circular cylinder is a typical bluff body and is one of the structural componentsmostly employed. Numerous investigations have been made of the flow past twocircular cylinders, which is the simplest case of group, in the last two decades. Anextensive review was given by Zdravkovich [1] and by Ohya et al. [2]. At subcriticalReynolds numbers, many interesting and unexpected phenomena were found, suchas the biased bistable flow which occurred at specific ratios between the two circularcylinders in a side-by-side arrangement and the fluid forces that yield discontinuitiesfor the two cylinders in tandem and staggered arrangements at certain spacing ratios.More detailed investigations of discontinuities of pressure distributions on twocircular cylinders in staggered arrangement were recently given by Gu and Sun [3,4].

Nevertheless, three circular cylinders in equilateral-triangular arrangement are thetypical case of cylinder group. It is extremely interesting to know whether theaerodynamic characteristics of two circular cylinders still exist or some new featuresmay happen due to the effect of the third one.

Gerhardt and Kramer [5] investigated the interference effects for groups of threeor four stacks. Eastop and Turner [6] studied three cylinders positioned in a straightline, either normal to or in the flow direction. Price and Paidoussis [7] studied themore general case of three cylinders in a staggered arrangement. They found, ingeneral, that the effect of cylinder displacement on the fluid forces for cylinder in agroup of three is very similar to that obtained with one cylinder in a group of two. Ithas been shown that, within a reasonable degree of accuracy, the force coefficients onone of the three cylinders may be calculated using a principle of superposition fromthe force coefficients on one cylinder in a group of two. Sayers [8] investigated thedrag and lift coefficients occurring on cylinder in a group of three equispacedcylinders. He concluded that the angle of orientation to the free stream will stronglyinfluence the force coefficients acting on any one cylinder and large reversals in liftmagnitude and direction occur at certain inclination angles. Lam and Cheung [9]studied phenomena of vortex shedding and flow interference of three cylinders indifferent equilateral arrangements, using a dye-injection technique flow visualizationat Reynolds numbers 2.1� 103 and 3.5� 103. Flow patterns were shot at 108intervals from 08 to 608. On the basis of wide and narrow wakes observed behind thecylinders, they concluded that the ‘bistable’ flow characteristic that existed at b ¼ 08for N=d ¼ 1:2722:29 depends very much on the starting conditions. Yunseok andChangkoon [10] reported the results of the close relationship between aerodynamicresponses and pressure distributions on three circular cylinders. More recently,Tatsuno et al. [11] studied effects of interference among three equidistantly arrangedcylinders in a uniform flow. The static pressures around the circumference of eachcylinder were measured by rotating the cluster from 08 to 608 in steps of 58. Theresults show that the effects of flow interference among the three cylinders are severewhen the spacing ratios are small. When two cylinders arranged side by side areupstream of or behind the third one the flow patterns are not symmetrical withrespect to the uniform flow direction, and the drag or lift coefficients of the two

Z.F. Gu, T.F. Sun / J. Wind Eng. Ind. Aerodyn. 89 (2001) 553–568554

cylinders in the side-by-side arrangement are not always equal to each other at smallspacing ratios.

This paper describes the pressure distributions on three equal circular cylindersarranged in an equilateral-triangular manner at different spacing ratios underdifferent angles of attack subjected to a cross uniform flow. Combined with smoke-wire technique flow visualization, more attention has been paid to the classificationsof flow patterns and the mechanisms of interference between cylinders.

2. Experimental apparatus and data reduction

It is believed that the different flow patterns could greatly affect the pressuredistributions on the surface of the cylinders, especially the fluctuating pressure; thus,the pressure measurement experiments were conducted first in a closed-return low-speed wind tunnel. The test section is 1.2m wide, 1m high and 8m long. Velocityvariation across the test section is � 1% and the turbulence intensity is 0.4%. Themaximum velocity is 36m/s. Two turntables are mounted on the floor of the testsection and they are located 1.5 and 5.5m from the entrance of the test section,respectively. The first turntable is usually used for experiments in uniform smoothflow and the second one for experiments in simulated atmospheric boundary layerwith the aid of triangle spires, barriers and roughness elements mounted from theentrance of the test section.

Each cylinder of the group is a 640mm long aluminum tube of 48mm externaldiameter, with a machine-finished surface. Pressure taps are provided every 108 atmid-span of the cylinders circumferentially. The cylinders with circular end plates of800mm diameter on both ends, were set up vertically in the upper and loweradjustable guide-slots. The lower guide-slots were fixed on the first turntable in thetest section. The whole assembly was then located at a distance of 1.5m downstreamof the tunnel throat.

The measurement system of the surface pressure consisted of pressure transducers(PDCR�23d), a set of Scanivalve (SGM�48), three DC amplifiers (6M72), an A/Dconverter and a personal computer (IBM PC/XT). Three individual transducers wereused for each cylinder such that the pressure signals at the same azimuth angle y (seeFig. 1) of three cylinders under test were recorded simultaneously. A typical run tookabout 60 s to complete the data logging of all pressure taps for three cylinders. Alltests were carried out at a wind speed of 18m/s with a Reynolds number of 5.5� 104

that is based on the diameter of a single cylinder.The configuration of three cylinders is shown in Fig. 1 with the sign conventions of

the fluid-force coefficients, in which d is the diameter of the cylinder, N, the distancebetween the centers of two cylinders, N=d, the spacing ratio between two cylinders,and b, the angle of incident flow. As frequent reference will be made to the individualcylinder, the upstream and the downstream cylinders, they are labeled as cylinders A,B and C, respectively, in later discussion. All the results of experiments wereexpressed in terms of dimensionless pressure coefficients. According to the usualpractice, the measured instantaneous pressure pðy; tÞ is expressed as the sum of

Z.F. Gu, T.F. Sun / J. Wind Eng. Ind. Aerodyn. 89 (2001) 553–568 555

time-mean pressure pðyÞ and the fluctuating pressure p0ðy; tÞ, where t is time. Time-mean and fluctuating pressure coefficients CpðyÞ and CprmsðyÞ are defined as Cp �ðyÞ ¼ ½ pðyÞ � p1=0:5rV2

1 and CprmsðyÞ ¼ spðyÞ=0:5rV21, where spðyÞ is the root-

mean-square value of p0ðy; tÞ, and p1, r V1 are the pressure, density and velocity ofoncoming flow, respectively. The drag and lift coefficients, denoted by CD and CL,respectively, are defined conventionally and were obtained by integrating appro-priately the pressure distribution around the circumference of each cylinder at midspan.

The experiments of flow visualization were performed in an open-circuit windtunnel with a working-section 6m long, 0.6m wide and 0.6m high. The wind flow inthis wind tunnel is generated by an axial fan located near the inlet, which is followedby a diffusion chamber and a nozzle leading to the working section. The maximumvelocity of free stream is about 36m/s with a background turbulence intensity of0.2%. The smoke-wire technique flow visualization was adopted. One of four wireswas placed in the front of the model and the other three wires were located behindthe model. Pictures were taken with a time-delay, between the generation of smokeand the flash of camera, of 80m/s. The Reynolds number, which is based on thediameter of a single circular cylinder for flow visualization experiment is l.3� 104.

3. Results and discussion

For perceiving the effect of group interference in the experiments of pressuremeasurement, the pressure distribution of a single cylinder was measured first andthen was compared with the results of three-cylinder experiments. The time-meanand fluctuating pressure distribution on a single cylinder at a Reynolds number of5.5� 104 are shown in Figs. 2(a) and 3(a), respectively. The time-mean andfluctuating pressure distribution on both sides of the cylinder are quite symmetrical

Fig. 1. Configurations of three cylinders in a group together with the sign conventions of the fluid-force

coefficients.


Fig. 2. (a–h) Time-mean pressure distributions on single and three circular cylinders in equilateral-

triangular arrangement (N=d ¼ 1:7) at Re=5.5� 104.


Fig. 3. (a–h) Fluctuating pressure distributions on single and three circular cylinders in equilateral-

triangular arrangement (N=d ¼ 1:7) at Re=5.5� 104.


and show a typical laminar-separation at about y ¼ �808. It is a typical pressuredistribution on circular cylinder at subcritical Reynolds number.

In the experiments of three cylinders, seven spacing ratios, i.e., N/d=1.7, 2.0, 2.2,2.5, 3.0, 4.0 and 5.0, were used out combined with angles of incident flow b variedfrom 08 to 608. To achieve the goal of this research, the increments or the steps of bwere not fixed, and they were carefully adjusted to appropriate degrees depending onthe change of pressure distribution on cylinders during the test.

As for the different levels of interference regarding the influence of spacing ratio,mainly four different affected regions can be identified. These are effects of small,transition, medium and large spacing.

3.1. Effect of small spacing: 1.74N/d42.0

For small spacing ratios, such as N=d ¼ 1:7, the interference between the threecylinders is very strong and complex. The characteristics of interference varysignificantly as the relative positions of cylinder vary with the angles of incident flow.In the most complicated case, the separated shear layers from upstream cylinder mayinteract directly with downstream cylinder at certain incident angles of flow.

According to the pressure distributions on downstream cylinder that varies withthe angle of incident flow, three basic types of interference with different essentialdistinctions can be classified. These are flow patterns of proximity effect, shear layersreattachment effect and wake effect. Furthermore, as the angles of incident flow varyfrom 08 to 608, seven steps of flow pattern can be subdivided. It was confirmed by theresults of flow visualization, which will be described in greater detail later on. Thetime-mean and fluctuating pressure distributions on the three cylinders at N=d ¼ 1:7are given for seven typical cases of incident flow in Figs. 2(b)–(h) and 3(b)–(h),respectively. The pictures of flow visualization corresponding to the seven typicalcases or steps are shown in Fig. 4, except for the case of b ¼ 08 (step 1), which will bediscussed below.

In order to figure out and better understand the classification in small spacing, theschematic illustration of the flow patterns for the various effects, including proximityeffect, shear layers reattachment effect and wake effect are given in Fig. 5.

3.1.1. Patterns of proximity effectThe characteristic of this type of effect is that the separated shear layers from each

cylinder, as well as their wakes, did not interact directly, at least in the near wake ofcylinders. Owing to difference in angles of incidence flow, two types of flow patternor pressure distribution can be distinguished.

(A) Symmetric flow pattern and pressure distribution: It is expected that this type offlow pattern and pressure distribution appears as the orientation of the group wassymmetric to the flow, i.e., b ¼ 08 and 608. The time-mean and fluctuating pressuredistributions on cylinders are shown in Fig. 2(b), (h) and Fig. 3(b), (h), respectively.Furthermore, based on the degree of stability of the pressure distributions oncylinders as the angle of incident flow changes, the stable or unstable pressurepattern may be identified.


At b ¼ 08, due to the blockage effect caused by the presence of downstreamcylinders, the base pressure of cylinder A reduces to �0.5, which is much less thanthat of the single cylinder case of �1. On the other hand, the minimum pressure onthe insides of both cylinders B and C increases to �1.5 due to the relatively higherspeed gap flow between the shear layers. The time-mean pressure distributions oncylinders B and C are almost symmetric, which agree with the results given byTatsuno et al. [11] in the case of N=d ¼ 1:73. The areas of positive pressuredistributions on both cylinders are reduced. The fluctuating pressure, which isexpressed as its root-mean-square, on the inner-side surfaces of both cylinders B andC is strong. It is caused by the proximity effect of the shear layers separated fromboth sides of cylinder A. It should be noted that there are only slight differencesbetween both the time-mean and fluctuating pressure coefficients on cylinders Band C.

During the tests, similar and rather stable symmetric pressure distributions wereobtained in different runs. It suggests, in other words, that there was no bistablewake phenomenon as mentioned by Lam and Cheung [9].

Fig. 4. Pictures of smoke-wire flow visualization for N=d ¼ 1:7 and Re=1.4� 104; (a) b ¼ 08; (b) b ¼ 0:58;(c) b ¼ 108; (d) b ¼ 158; (e) b ¼ 338; (f) b ¼ 368; and (g) b ¼ 608.


In contrast with the results of time-mean pressure distribution, the results ofsmoke-wire flow visualization at Reynolds number l.3� 104 show unstablecharacteristics of the flow property in this case. Three typical pictures can be pickedout in the case of b ¼ 08. They show symmetric flow pattern with respect to the free-stream direction and (Fig. 4(a) case 1) asymmetrical flow patterns with the biased

Fig. 5. Schematic illustration of the flow patterns for the various effects, including proximity effect, shear

layers reattachment effect and wake effect in small spacing.


wake-flow either switching to cylinder B (Fig. 4(a) case 2) or C (Fig. 4(a) case 3),respectively. It is somewhat in accord with the finding of Lam and Cheung [9], whoused dye-injection technique flow visualization to show the bistable flow appearingin the range 1:275N=d52:29. The three types of flow pattern pictures caught wererather stochastic other than the characteristic of bistability as described in Ref. [9].The results obtained by Tatsuno et al. [11] confirmed that the bistable flowphenomenon happened at N=d ¼ 1:39 and 1.73 and showed that the pictures of flowpatterns are asymmetrical with respect to the free-stream direction at Reynoldsnumber 507 (only one case). However, Tatsuno et al. [11] indicated that in the case ofN=d ¼ 1:73, there are slight differences in pressure coefficient distributions betweencylinders B and C at Reynolds number 6.2� 104.

As mentioned by Tatsuno et al. [11], the bistable flow is always associated with thecylinder with a narrow wake and experiences large drag and lift forces in comparisonwith the cylinder with a wider wake. Zdravkovich and Pridden [12] described that theintermittence of the high- and low-drag values did not cease but persisted for alonger time at one value. Lam and Cheung [9] also reported that whether the largerwake forms behind cylinder B or C depends on the starting conditions and that oncewide or narrow wake has been established behind a given cylinder then it remains inthis pattern. Therefore, in the authors’ opinion, the phenomenon of biased bistableflow could be detected by the measurement of time-mean pressure and in this casethe phenomenon may be classified as a very unstable gap flow. More detailedinvestigation of three cylinders in this phenomenon combined with the effect ofReynolds number should be carried out.

It should be emphasized that this type of symmetric pressure distribution onlyappears at very small region around b ¼ 08. As the wind angle changes a little bit,such as b ¼ 0:58 which will be discussed in detail later, the symmetric pressuredistribution no longer exists. Thus, this type of pressure distribution is identified asan unstable or critical symmetric pressure distribution because it is highly sensitive tothe changing of the angles of incident flow.

At b ¼ 608, the symmetric flow pattern (Fig. 4(g)), as well as pressure distributions(Figs. 2(h) and 3(h)) with respect to the free-stream direction appear again due to thesymmetric geometrical arrangement of the three cylinders.

The stagnation point on upstream cylinders A and B turns to their inner-sideabout 108, respectively, and the area of positive pressure distribution is about58 wider than that of the single cylinder. The positive pressure area on cylinder C,on the other hand, becomes much narrower. The base pressures of three cylinders arealmost the same but higher than that of a single cylinder. The fluctuating pressure isstronger than that of a single cylinder and the values of fluctuating pressurecoefficients are greater than 0.3 on most surface areas of the cylinder. At b ¼ 608, thepicture of flow visualization shows that individual wake structures formed behindeach cylinder. Due to the presence of cylinder C, the wakes of cylinders A and Binclined outward, whereas an extremely wide wake takes shape behind cylinder C.The symmetric pressure distribution is maintained in a rather wide area of angle ofincident flow and could be observed up to b ¼ 408. These features are significantlydifferent from the case of b ¼ 08. In other words, this type of pressure distribution is


not sensitive to the change in angles of incident flow. It, therefore, may be attributedto the stable symmetric pressure distribution.

(B) Asymmetric distribution: At a certain region adjacent to the stable symmetricflow pattern or pressure distribution area, e.g. b ¼ 368, the flow around each cylinderretains its individual complete wake structure (see Fig. 4(f)). However, there is stillno direct interference between the separated shear layers, as well as the wakes, atleast in the near wake of the cylinders. It is only the serious effect of proximity, whichcauses the pressure distribution to be asymmetric (Figs. 2(g) and 3(g)). Theinterference between cylinders, in this case, causes the streamlines to be squeezedand distorted.

3.1.2. Patterns of shear layer reattachment effectThe characteristic of this type of effect is that the shear layer separated from

upstream cylinder reattaches on either the outside or inside surface of thedownstream cylinder.

At the boundary regions of unstable or stable symmetric pressure distributions,such as at b ¼ 0:58 or 338, the shear layer separated from cylinder A reattaches onthe inside or outside of cylinder C, respectively (Figs. 4(b) and 4(e)). Due to thereattachment of the shear layer with relatively high velocity and vorticity in it, thepressure distributions change dramatically on cylinder C (see Fig. 2(c) and (f)). Alarge area of strong suction pressure forms. The peak value fluctuating pressurecoefficients may reach 0.4 (Fig. 3(c) and (f)). A rather strong fluctuating lateral forceresults, which tends towards the centerline of the group in the case of b ¼ 0:58, oraway from it in the case of b ¼ 338. During the test, a hot wire probe was placed atthe near-surface region, where a strong suction pressure was present. The signal ofvelocity fluctuation was sampled and processed later. A peak frequency could bedetected in the power spectrum analyses, which corresponds to a Strouhal number of0.4. It is suggested that the appearance of the great suction on one side of cylinder Cis associated with some regular vortex shedding with rather high frequency. It issomewhat similar to the case of two circular cylinders in staggered arrangement (e.g.see Refs. [3,4]).

Due to the changes in orientation of incident flow, the shear layer separated fromcylinder A is somewhat away from cylinder B, the pressure distribution on cylinder Bapproaches that of the single-cylinder case.

3.1.3. Patterns of wake effectFor the orientations of incident flow between these two regions of shear layer

reattachment effect, the flow patterns may be classified as wake effect. Thecharacteristic of this type of effect is that cylinder C is entirely or partially submergedin the wake of cylinder A (Fig. 4(d) and (e)), which corresponds to the time-meanpressure distribution on cylinder C with double-peak (at b ¼ 108, Fig. 2(d)) or single-peak (at b ¼ 158, Fig. 2(e)), respectively. On the other hand, the fluctuating pressuredistribution on cylinder C reveals four or two peaks respectively (Fig 3(d) and (e)).

At b ¼ 108, particularly, the double-peak pressure distribution pattern on cylinderC is almost symmetric. It is very similar to the case of downstream cylinder of two


circular cylinders in tandem arrangement, except that the symmetric line shifts about308. The angle between the incident flow and the centerline of cylinders A and C is208. It could be interpreted as the proximity effect of cylinder B that causes the wakeof cylinder A to decline about 208.

Yunseok and Changkoon [10] schematically drew six steps of flow pattern as theangles of incident flow vary from 08 to 608. However, according to the presentexperiments, step 4 corresponds to the time-mean pressure distribution of single-peak on cylinder C, free-stream could reach the surface of cylinder C directly as it isonly partially submerged in the wake of cylinder A.

Fig. 6(a) shows the drag and lift coefficients on each cylinder as the group isrotated through 608 about its axis in small spacing N=d ¼ 1:7.

The values of drag coefficient CD of cylinders A and B are close to each other andincrease slowly from 0.8 at b ¼ 08 to 1.2 at b ¼ 608. The value of CD for cylinder Cdecreases from 1.0 at b ¼ 08 to its minimum value of �0.2 at b ¼ 258 and thenincreases. At b ¼ 608, it is slightly less than that of cylinders A and B.

The lift force coefficients CL on both cylinders A and B are small. As b > 208, theCL of cylinder B increases slightly as the angle of incident flow increases, the reverseis true for cylinder A, and they reach values of � 0.15 at b ¼ 608, respectively.

On the other hand, CL on cylinder C changes significantly; particularly at small b,such as b ¼ 0:58, CL may reach a value of +0.8, but it drops to �0.2 abruptly atb¼ 7288, and then it maintains the same value up to b ¼ 148. It drops again andreaches its minimum value of �1.3 at b ¼ 258, and then recovers gradually up to zeroat b ¼ 608.

In summary, the CD values of cylinders A and B have no significant difference andalso have no significant changes with various angles of incident flow. The significantchanges in drag and lift forces on cylinder C are caused by different effects ofinterference. It is interesting that both the minimum drag and lift forces (theirabsolute value) on cylinder C occur almost at the same angle of b ¼ 258; thus, asignificant outward force results. At small values of b (b5108), a significant resultantforce results, which is combined with rather strong lift (CL ¼ 0:8) and drag(CD ¼ 1:0) forces. The effect of shear layer reattachment is the most serious case ofinterference between three cylinders.

3.2. Effect of medium spacing: 2.54N/d43.0

As the spacing ratio increases to a certain value, such as N=d ¼ 2:5, the pressuredistributions on cylinders show rather different features as compared with the case ofsmall spacing ratios. At the medium spacing ratio, the shear layer separated fromcylinder A no longer acts on cylinder C directly and the effect of shear-layerreattachment pressure distribution no longer exists. However, the suction area,which is weaker than the case of shear layer reattachment in small spacing ratios, isstill present on the outer side of cylinder C; it is caused by the gap flow of relativelyhigh velocity adjacent to the separated shear layer of cylinder A.

The symmetric type of pressure patterns also can be expected to occur at b ¼ 08and 608; however, they reveal that the effect of proximity reduces greatly. In the


Fig. 6. (a–d) Drag and lift forces on each of the three cylinders in equilateral-triangular arrangement at

different spacing ratios.


region of wake effect, because of the increment of spacing ratios, the wake behindupstream cylinder could be developed more sufficiently; the wake influence oncylinder C is stronger than the case in small spacing with respect to the fluctuatingpressure.

The CD values of cylinders A and B are quite different (Fig. 6(c)). The values ofCD of cylinder B are about 0.4 larger than those of cylinder A regardless of changesin the angles of incident flow until the vicinity of b ¼ 608, where their difference isreduced. As the spacing ratio increases further, the tendency of reducing theirdifference shifts to the early angle of incident flow.

The curve of CD of cylinder C is similar to those in small spacing ratios; however,the minimum CD of negative value no longer exists. The CL values of cylinders Aand B are also similar to those of small spacing ratios, whereas, the CL of cylinderC changes and the location of strong positive lift force shifts to b > 148. Thevariation of lift force on cylinder C with the angles of incident flow forms a curve ofone peak and one valley with a smooth area between them. The location of thesmooth area coincides with the location of the area of minimum values for the dragcurve.

3.3. Effect of transition spacing: N/d=2.2

Between the small spacing and the medium spacing is the transition spacing(N=d ¼ 2:2). Due to the space limitations of the paper, the changes of pressuredistributions on cylinders are not shown and discussed here. The drag curves ofthree cylinders remain unchanged as those of small spacing ratios (Fig. 6(b),whereas, the lift curves, particularly CL of cylinder C show a transition processbetween the curves of small (N=d ¼ 2:0) and medium (N=d ¼ 2:5) spacing ratios.

3.4. Effect of large spacing: N/d54.0

As the spacing ratio increases up to or larger than 4, the interference on the time-mean pressure distribution is reduced further, but the reverse is true for thefluctuating pressure on cylinders. It could be explained by the re-establishment of theregular vortex shedding behind cylinders A and B; therefore, most interference oncylinder C is caused by the effect of wake.

The curves of drag force are similar to those at medium spacing ratios but aremore smooth (Fig. 6(d)). At the vicinity of b ¼ 608, the values of CD of the threecylinders are closer to each other. The CL curves of cylinders A and B are rathersymmetric. The CL curve of cylinder C changes gradually and looks like oneperiod of sine wave, which is similar to or agrees well with the result reported byTatsuno et al. [11].

The contours of CD and CL of cylinder C are presented in Fig. 7. It is shown that,within the medium and large spacing regions, both the drag and lift coefficients oftheir minimum values (the absolute value) are located at the same area ofb ¼ 202308.


4. Conclusions

The main results may be summarized as follows:(1) The angles of incident flow strongly influence the pressure distributions on any

one of the cylinders.(2) Due to the interference of the shear layers and/or wakes, the most complex

features of pressure distributions can be found on the downstream cylinder.(3) As regards to different levels of interference of the influence of spacing ratio,

mainly four different effect regions can be identified. These are effects of small,transition, medium and large spacing ratios.

(4) Three basic types of interference between three cylinders and therefore the flowpatterns can be classified because of the various angles of incident flow in small

Fig. 7. Contours of CD and CL of cylinder C with various N=d and b:


spacing ratios. These are flow patterns of proximity effect, shear layer reattachmenteffect and wake effect. Furthermore, seven flow patterns or steps can be subdividedas the angles of incident flow vary from 08 to 608.

(5) The effect of shear layer reattachment on the downstream cylinder, which is themost serious interference case between cylinders, yields a significant lateral forcedirected either inward or outward from the center of the cylinder group dependingon the orientations of incident flow. This phenomenon is associated with some kindof regular vortex shedding.

Acknowledgements

The project is supported by the Research Fund for the Doctoral Program ofHigher Education (China). The authors thank to the anonymous reviewers for theiruseful comments and suggestions. The authors are grateful to Professor Ren Wangfor his great help in preparing the revised manuscript and to Mr. Xiangdong Zhaofor his assistance in the flow visualization experiment.

References

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Documents

Classifications of Flow Pattern on Three Circular Cylinders in Equilateral-triangular Arrangement