Selected Paper

7/31/2019 Selected Paper

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An experimental study on the cross-ow vibration of a exible cylinder in cylinder arrays

Tsun-kuo Lin, Ming-huei Yu *

Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, ROC

Received 31 October 2003; accepted 26 June 2004

Abstract

In the experiment, a monitored cylinder equipped with two accelerometers inside was exibly mounted in a water tunnel, sur-rounded by one to six identical cylinders elastically mounted in rotated triangular pattern. The amplitude diagrams, spectra andorbits of the cylinder motion are used to examine the vibration behavior of the cylinder under the various test conditions of thefree stream velocity, the number of the surrounding cylinders, and the cylinder s natural frequency. In the case of the monitoredcylinder having the same natural frequency as the surrounding cylinders (22 Hz), amplitude response shows that uid elastic insta-bility occurs when the ow velocity is above a critical value for the cylinder in all the six-cylinder arrays. Above the critical velocity,the cylinder vibrates around an oval orbit with line-dominated spectrum, implying that the cylinder behaves like an oscillatorwith the streamwise and cross-stream responses have the same frequency but with a phase shift. By comparison of amplitude dia-grams of the cylinder in the six bundles, it reveals that the upstream cylinders have signicant inuence on the amplitude response of the monitored cylinderpromote the uid elastic instability of the monitored cylinder, and enhance the cylinder vibration above thecritical velocity. The downstream cylinders could suppress the vibration amplitude while the number of the downstream cylinders

has little effect on the amplitude response. In case of the monitored cylinder having different natural frequency from that of the sur-rounding cylinders, it is found that the difference in natural frequency of the cylinders has little effect on the critical velocity, butstrong inuence on the vibration amplitude above the critical velocity. 2004 Elsevier Inc. All rights reserved.

Keywords: Flow-induced vibration; Circular cylinder; Cross ow

1. Introduction

Bundles of circular cylinders are commonly used inengineering applications as in power transmission lines,

pipelines in deep water, suspension bridges, heatexchangers, etc. Their potential vibrations induced byuid ow have been extensively studied for the last threedecades. Blevins [1] summarized the most important re-sults of the ow-induced vibrations of circular cylinders.Later, researches on the ow-induced vibration of two

identical circular cylinders was continued because of itsfundamental importance, by Matsumoto et al. [2], andDielen and Ruscheweyh [3], for instances. Meanwhile,many researchers, Pettigrew and Taylor [4] and Schroe-

der and Gelbe [5], among others, focused on the uid elas-tic instability of cylinder rows and arrays because of theirpractical applications. While the ow-induced vibrationsof two cylinders and cylinder arrays are the main focusesin the previous studies, the ow-induced vibrations fromtwo cylinders to cylinder arrays are systematically inves-tigated in the study, an effort to bridge the gap betweenthe two cases in aspect of ow-induced vibration.

For cylinder bundles in cross ow, uid elasticinstability is an excitation mechanism for ow-induced

0894-1777/$ - see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.expthermusci.2004.06.004

* Corresponding author. Tel.: +886 7 5252000x4238; fax: +886 75254299.

E-mail address: [email protected] (M.-h. Yu).

www.elsevier.com/locate/etfs

Experimental Thermal and Fluid Science 29 (2005) 523536
mailto:[email protected]:[email protected]


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vibrations, resulting in large-amplitude vibrations andeven failure of the cylinder bundles. Fluid elastic insta-bility, which may occur in multiple circular cylinders,is a result of the interaction between the cylinders andthe owing uid. In cylinder bundles, the interactionsof the cylinders with the uid, and the coupling amongthe cylinders through the uid are very complex. How-ever, it is reasonable to suppose that the unsteady uiddynamic forces on a cylinder are mainly induced by

vibrations of the cylinder itself and its neighboring cylin-ders [6,7]. With this assumption, the vibration behaviorof a cylinder in cylinder bundle should be most affectedby its surrounding cylinders. For instance, a cylinder intwin-cylinder bundle may have different vibrationbehavior from the cylinder surrounded by more cylin-ders under otherwise the same condition.

The paper is aimed to describe the ow-inducedvibrations of a exibly mounted cylinder surroundedby various numbers of cylinders in cross ow. Theemphases of the study are the effects of the number of surrounding cylinders, and the natural frequency of the cylinder on the vibration behavior of the cylinder.For these purposes, the amplitudes, orbits, and spectraof the cylinder vibration are obtained by simultaneousmeasurements of the cylinder vibration in the stream-wise (X ) and lateral ( Y ) directions.

2. Experimental aspects

2.1. Water tunnel and instrumentation

A low speed water tunnel was designed and built forthis study, as shown schematically in Fig. 1 . The length

of the contraction section is approximately 0.6m and thecontraction ratio is 9:1. The test section, made up of bolted acrylic plates, has internal dimensions of 0.2m 0.2m. A tank with 0.76m 3 capacity serves asan accumulator placed after the test section. The tem-perature of the water ow was at 20 C. In operatingcondition, water ow was driven by a variable speedpump. After passing through a lter, the water entereda diffuser, which provides a gradual transition from 10

cm diameter circular pipe to a rectangular channel of 0.6m 0.6m. Before the 9:1 contraction, a layer of hon-eycomb and several screens were installed in the rectan-gular channel to reduce turbulence. At the entrance of the test section, a pitot tube was used to monitor the in-let ow velocity. The average water velocity in the testsection of the facility was up to 0.74m/s. The Reynoldsnumber, based on the inlet ow velocity and the diame-ter of test cylinders is 1.1 104 . The average upstreamturbulence intensity over the ow range is about 1.2%measured by hot-lm anemometry. The velocity distri-bution at the test section, excluding the boundary layer,is found to be at within 1%.

In the test section, stainless steel tubes were used toform cylinder bundles for testing. Six cylinder bundleswere tested in the experiment, they all being in a 60 equilateral triangle pattern. As shown in Fig. 2 (a), eachbundle consists of a monitored cylinder (bold circles inthe gure), equipped with two accelerators inside, andone to six surrounding cylinders. The cylinder bundlesare named as Array IVI, respectively, according tothe number of the surrounding cylinders. The diameterof the entire cylinders is 16mm, and the ratio of the cyl-inder center-to-center spacing to cylinder diameter P /Dis 1.33. Fig. 2 (b) is a side view of the test section to show

Nomenclature

Arms r.m.s. value of tube vibration amplitude (m)Ax tube vibration amplitude in the streamwise

(X ) direction (m)

A y tube vibration amplitude in the lateral ( Y )direction (m)c damping coefficientD outer tube diameter (m)E voltage output from the measurement system

(Volt)F force per unit length (N/m) f n the natural frequency of the monitored cylin-

der (Hz) f s the natural frequency of the surrounding cyl-

inders (Hz)m mass per unit length (kg/m)P pitch (m)Re Reynolds number

St Strouhal numberU crit critical velocity (m/s)U in inlet velocity (m/s)

x displacement in the streamwise ( X ) direction(m)x rms the r.m.s. value of the cylinder displacement

in the X -direction (m)X , Y rectangular Cartesian coordinates y displacement in the lateral ( Y ) direction (m) yrms the r.m.s. value of the cylinder displacement

in the Y -direction (m)

Greek symbolsf damping factor (dimensionless)l dynamic viscosity (kg/ms)m kinematic viscosity (m 2 /s)q density (kg/m 3 )

524 T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523536


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how to mount a cylinder in the test section. Each stain-less tube in the test array is suspended by two tensionalsteel wires at both ends. The tension in the steel wire canbe adjusted to change the natural frequency of the cylin-der. As shown in the gure, a stainless cylinder 198mmlong, 16mm outside diameter and 1.5 mm wall thicknessis mounted on a 1.2 mm diameter steel wire. The weightof each cylinder is 153g within 1% deviation. The freelength of the 1.2mm diameter wire is 180mm and thescrew-nut adjustments at both ends of the wire provide

ne tension adjustment for tuning the natural frequency.In the experiment, the natural frequencies of the cylin-ders surrounding the monitored cylinder were all xedat f s = 22Hz, and the natural frequency of the moni-tored cylinder, f n , was set at 22 Hz, 16.5Hz (=3/4 f s ),11Hz (=1/2 f s), and 27.5Hz (=5/4 f s ) for tests with differ-ent natural frequencies.

The instrumentation includes two accelerometersmanufactured by Endevco (Model #25A) with sensitiv-ity 4.693mV/g and frequency response of 28kHz. Thetwo accelerometers were installed inside the tube at mid-span. Their orientations were adjusted such that theywere sensitive to tube vibrations only in the streamwise(X ) and lateral ( Y ) directions, respectively. The outputof each accelerometers was amplied by a B&K chargeamplier (Model #2693) and coupled to B&K doubleintegrator circuit (Model #0788), as shown in Fig. 3 .The double integrator circuit, consisting of an input l-ter and two operational amplier integrators, was usedto integrate an acceleration signal twice and providean output signal proportional to displacement. The X -and Y displacement signals obtained from the doubleintegrators were digitized simultaneously at a samplingrate of 512 data/s by a 16 bit IOtech IEEE 488 A/D con-verter (ADC488/8SA) and transmitted to a PC compu-

ter for data storage and further processing. An FFTanalyzer, AND AD3524, was connected to the doubleintegrators to monitor the amplitude spectra of thecylinder during the experiments.

The voltage outputs from the data acquisition systemwere calibrated against the vibration amplitudes beforevibration measurements. In calibration, the tube withthe two accelerometers inside was given a circular mo-tion. Typical time series of displacements in the X andY direction are shown in Fig. 4 (a). The amplitudes of

the sinusoidal output voltages for the X - and Y -direc-tions were recorded and then plotted against the vibra-tion amplitude (the radius of the circular motion). Asshown in Fig. 4 (b), the calibration data for both theX - and Y -directions are tted to a straight lineE = aA + b, where A is the vibration amplitude and E is the voltage output. Curve-tting gives the values of the coefficients, a = 0.10 and b = 0.007. It is thus shownthat the voltage outputs can linearly represent the vibra-tion amplitudes in the experimental range.

2.2. Uncertainty

To characterize the magnitude of cylinder vibration,the r.m.s. value of vibration amplitude A rms is intro-duced in dimensionless form as,

Arms = D ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2rms y 2rmsq = D 1where D is the diameter of the cylinder. x rms and yrmsare the r.m.s. values of the cylinder displacements inthe X - and Y -directions, respectively.

The displacements were obtained by digitizing theoutput voltages from the measurement system thattransforms the displacements into voltage signals. The

Flow

Filter

ScreenHoneycomb

Tank

Test section

10 h.p.motor

Regulator

Speed control unit

220 V a.c. supply

Pitot tube

Fig. 1. Water tunnel facility.

T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523536 525


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uncertainties of the displacement measurements couldbe due to the accuracy of the vibration measurementsystem, and of the data acquisition system. The resolu-tion of the data acquisition used to digitize the voltagefrom the double integrator circuit is 4 10

4 V i n arange of 5V. The tolerance of the measured amplitudedue to the measurement system is estimated to be0.3mm, based on the standard deviation of the calibra-tion data from the tting curve, shown in Fig. 4 (b).

The velocity prole in the center region of the testsection, exclusive of the boundary layer region, is uni-form within 5% variation at the maximum freestreamvelocity U in = 0.74m/s, and the turbulence intensity is1.3%, measured by a Dantec hot-lm anemometer.

The tolerance in measuring the natural frequency is

0.2Hz in the range of 931Hz. The natural frequencyand damping factor of a cylinder were determined bymeasuring the amplitude response of the cylinder aftergiven a perturbation in otherwise still water, as shownin Fig. 5 for a typical amplitude response in time do-main. The damping factor can be estimated from thedecay of the amplitude, and the natural frequency of the cylinder can be estimated by spectral analysis of the time trace. For each natural frequency, the averagevalue from several measurements is adopted, and theuncertainty can be estimated by evaluating the standarddeviation. Perturbations of various directions were ap-plied in the measurements to make sure that the naturalfrequency is independent of cylinder vibration orienta-tion in the experimental setup.

For spectral analysis of the vibrations of the cylinderin cross ow, the software MATLAB was used to obtainspectra data. Additionally, a spectrum analyzer ANDAD3524, with a setting of frequency resolutionD f = 0.125Hz, was connected to the voltage output tomonitor the cylinder vibration during the experiments.

The uncertainty of mass damping can be estimatedby using Moffat s [8] uncertainty analysis. The uncer-tainties of the experimental parameters are summarizedin Table 1 .

3. Results and discussion

3.1. Vibration amplitude

For the effect of surrounding cylinders on the vibra-tion of the monitored cylinder, vibration measurementsof the monitored cylinder were carried out with the sur-rounding cylinder added one by one, while the naturalfrequency of the entire cylinders was set at the same(22Hz). Figs. 6 (af) show the amplitude responses of the monitored cylinder in the six-cylinder arrays,respectively. In each of the gures, there are two curvesrepresenting the vibration amplitudes in the X - and Y -directions. It is seen that all the amplitude curves havesimilar trend as the ow velocity is increased. The vibra-tion amplitudes are very small, slightly increasing withthe ow velocity, as the velocity is below a critical value U in % 0.5m/s for all the test arrays except U in % 0.6m/sfor the two-cylinder array. These small-amplitude vibra-tions could be attributable to turbulence, consideringthat the monitored was in the wake region behind theupstream cylinders. Above the critical velocity, thevibration amplitudes increase signicantly as the ow

Fig. 2. (a) Top view of the test cylinders. (b) Side view of the cylindersin the test section.



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velocity is increased, with the amplitude in the Y -direc-tion being larger than that in the X -direction for each

test case. The continuous increase of vibration ampli-tude with ow velocity implies that uid elastic instabil-ity has occurred.

The serious vibration above the critical velocity couldbe explained as follows. When the ow velocity is highenough, the uid has sufficient energy to excite the cylin-ders to vibrate with certain amplitude at their naturalfrequency. Through the coupling of the uid surround-ing the cylinders, resonance occurs among the cylindersthat have the same natural frequency, and as a resultthe vibration amplitude was amplied and thus seriousvibration was observed. The explanation of the seriousvibration due to uid elastic instability in the viewpointof resonance is consistent with the experimental ndingsthat the dominant frequency of the uid ow around thecylinders is the natural frequency of the cylinders. Asshown in Fig. 7 are typical time history of the ow aroundthe cylinders and the corresponding velocity spectrum,obtained by using a Dantec hot-lm anemometer

X directionAcceleration

Meter

Y directionAcceleration

Meter

Amplifierand

Double IntegratorCircuit

ComputerFFT Analyzer

DataAcquisition

System

Fig. 3. The data acquisition system for 2-channel vibration measurements.

15 20 25 30 35 40

1.5

2.0

2.5

3.0

3.5EyEx

E x ,

E y (

V o l

t )

Vibration amplitude (mm)

-20

-10

0

10

20

0.0 0.5 1.0 1.5 2.0 2.5

-20

-10

0

10

20

D i s p l a c e m e n

t ( m m

)

Time(sec)

Ay

Ax

(a)

(b)

Fig. 4. (a) Typical time series of displacements in the X and Y direction, (b) the calibration of the output voltages from thedata acquisition system versus the vibration amplitudes, E x for theX -direction, and E y for the Y direction.

0 2 3

-2

-1

0

1

2

y / D ( % )

t(sec)1 4 5

Fig. 5. A typical time trace of the cylinder vibration after perturbationin otherwise still water.



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(StreamLine) with a hot-lm probe (type 55R11). Simi-lar spectra were obtained regardless of the inlet owvelocity once the velocity is high enough. The velocity

spectra indeed exhibit only one spectral peak at f n(22Hz), indicating that the uid ow oscillates at thenatural frequency of the cylinders. The oscillation fre-quency is not the vortex shedding frequency that is lin-early related to ow velocity. The Strouhal number of the vortex shedding of the two-cylinder conguration,for instance, is estimated to be 0.12 (by the gure inRef. [1, p. 176]). The vortex shedding frequency in theow velocity range, U in = 0.60.7m/s, is then about5Hz, not 22Hz. Due to the mismatch of the two fre-quencies, vortex-induced instability is not likely respon-sible for the large-amplitude vibration in the velocityrange. The cylinder vibration is identied as a result of uid elastic instability, based on the character of thevibration amplitudes that increase rapidly once a criticalcross-ow velocity is exceeded, as shown in Fig. 6 . It isknown that the vibration amplitude of uid elasticvibration increases very rapidly with the ow velocityonce a critical cross-ow velocity is exceeded, while theamplitude response of a vortex-induced vibration is typ-ically a hump, or multiple humps if vibrations also occurat sub and super harmonics of the shedding frequency(see Chapters 3 and 5, by Blevins [1], for instance). Withthe uid ow and the cylinders oscillate dominantly atthe natural frequency, almost independent of the inlet

ow velocity, the resonance of the cylinders throughthe uid coupling is evident for the case that all the cyl-inders have the same natural frequency. The effect of thenatural frequency difference between the monitored cyl-inder and its surrounding cylinders on the cylindervibration will be further discussed in Section 3.3.

It is noted that before the amplitude response beingused to determine the critical velocity, the amplitude velocity curve should be checked if hysteresis exists.For this purpose, the vibration amplitude in both theX - and Y -directions were examined with the ow veloc-ity being increased from zero to 0.74m/s and thendecreased to zero, as shown in Fig. 8 for the seven-cylinder array, as an example. No obvious hysteresiswas observed for all the cylinder arrays in the experi-mental range. The experimental results of vibrationamplitude are also compared with the experimental re-sult by Kassera and Strohmeier [9], as shown in Fig. 9 .The test conditions and results of the two studies aresummarized in Table 2 . From the comparison, it isshown that the experimental results agree reasonablywith previous studies at least in low velocity range. Afterthese checks, the amplitudevelocity curves are thenused to determine the critical velocity for the onset of the uid elastic instability of the cylinder. The criticalow velocity here is dened as the ow velocity where

0.0 0.2 0.4 0.6 0.81 1.0-0.004

-0.003

-0.002-0.001

0.000

0.001

0.002

0.003

0.004

E ( v o l

t )

Time(sec)

0 10 20 30 40 50 60 70 80 90 100-0.002

0.000

0.002

0.004

E ( v o l

t )

Uin=0.5 m/s

f(Hz)

(a)

(b)

Fig. 7. (a) Typical time history of the uid velocity around the cylinder(b) the corresponding velocity spectrum.

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-202468

1012141618202224

velocity increasevelocity decrease

f n=f s

x r m s

/ D ( % )

Uin(m/s)

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-202468

1012141618202224 velocity increase

velocity decreasef n=f s

y r m s / D

( % )

Uin(m/s)

Uin /fnD

Flow

(a)

(b)

0 0.5 1.0 1.5 2.0

Fig. 8. Vibration amplitudes of the monitored cylinder in the seven-cylinder array with ow increasing from 0 to 0.74m/s and thendecreasing.



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the vibration amplitude exhibits sudden increase, fol-lowing the criterion used by Weaver and El-Kashlan[10].

The curves of the dimensionless amplitude Arms /D asfunction of inlet velocity are shown in Fig. 10 . With oneupstream cylinder, the monitored cylinder in Array I hassudden increase of vibration amplitude at the freestream velocity U in $ 0.6m/s, equivalently the reducedvelocity about 1.7. With one more upstream cylinder(Array II), the reduced velocity decreased to about 1.5.It is therefore illustrated that with two upstream neigh-boring cylinders, the cylinder in Array II becomes moreelastically unstable than that in Array I. In Array I,there is no gap ow between two cylinders impingingon the monitored cylinder, the unique ow situationmay be the reason that the ow velocity is required tobe at a higher value for onset of uid elastic instabilityfor Array I compared to Array II, and other arrays.With three upstream cylinders (Array III), the criticalvelocity is approximately the same as Array II. How-ever, the vibration amplitude above the critical velocityis larger in Array III than Array II, implying that moreupstream cylinders provide more and better coupling

among the cylinders once above the critical velocity.The resonance among the cylinders is therefore en-hanced, given the condition that all cylinders are atthe same natural frequency. Thus signicant vibrationamplitude was detected for the cylinder in Array III.With one cylinder added downstream (Array IV), thecritical velocity remains the same, but the vibrationamplitude above the critical velocity is suppressed, back

to the case of Array II approximately. With one or twomore downstream cylinders (Array V, VI), the criticalvelocity and vibration amplitude beyond the criticalvelocity are basically the same.

In summary, in case all the cylinders having the samenatural frequency, the upstream cylinders have signi-cant inuence on the amplitude response of the moni-tored cylinder; either promote the instability byforming a gap ow impinging on the monitored cylin-der, or enhance the cylinder vibration through betteruid coupling with more upstream cylinders. The down-stream neighboring cylinders could suppress the vibra-tion amplitude while the number of the downstreamcylinders has little effect on the amplitude response forthe present conguration.

3.2. Orbit of cylinder vibration

The vibration behavior of the cylinder can be furtherunderstood by examining the orbit of the cylinder mo-tion in the uid ow. Fig. 11 shows the orbits of themonitored cylinder in the six test arrays. The orbits wereplotted by tracing out the time-dependent displacementsof the cylinder motion in the X - and Y -directions forduration of 16s.

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-202468

1012141618202224

cylinder array Icylinder array IIcylinder array IIIcylinder array IVcylinder array Vcylinder array VI

f n =f s

A r m s /

D ( % )

Uin(m/s)

Uin /fn D0 0.5 1.0 1.5 2.0

Fig. 10. Vibration amplitudes of the monitored cylinder in differentcylinder arrays. In the case of f n = f s , i.e. all the cylinders have the samenatural frequency.

0.0 0.5 1.0 1.5 2.0 2.5-202468

10

12141618202224 present study(exp. data)

Kassera and Strohmeier(exp. data)

A r m s / D

( % )

Uin /f

nD

Fig. 9. Comparison of the experimental results of vibration amplitudewith the experimental data by Kassera and Strohmeier [9]. Theexperimental and computational conditions are listed in Table 2 .

Table 2Test conditions of the present experiment, and the experiment byKassera and Strohmeier [9]

Investigators The present authors Kassera andStrohmeier [9]

P /D 1.33 1.2D (mm) 16 16Arrangement 60 (rotated triangular) 30 (triangular)Tube location Central row Central rowf 0.0096 0.0145m(2pf )/qD 2 0.19 0.225 f n (Hz) 22 27.2



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At the low velocity U in = 0.28m/s, less than the criti-cal velocity, the cylinder in all the six arrays vibratesrandomly around a xed point. The small amplitudeand random nature of the orbits illustrate that the cylin-der vibration in the low velocity range is due to turbu-

lence. At the ow velocity slightly above the criticalvalue, such as U in = 0.55, and 0.60m/s, the orbits looklike an ellipse. The elliptical orbits imply that thestreamwise and cross-stream responses have the samefrequency but with a phase shift. The cylinder obviously

(a)

U in=0.28 m/s 0.55 0.60 0.64 0.68 0.74

25% D

25% D

(b)

U in=0.28 m/s 0.55 0.60 0.64 0.680.74

(c)

U in=0.28 m/s 0.550.53 0.57 0.60

(d)

U in=0.28 m/s 0.55 0.60 0.64 0.68 0.74

(e)

U in=0.28 m/s 0.55 0.60 0.64 0.68 0.74

(f)

U in=0.28 m/s 0.55 0.60 0.64 0.68 0.74

0.64

Fig. 11. Orbits of the monitored cylinder in the six-cylinder arrays: (a) Array I, (b) Array II, (c) Array III, (d) Array IV, (e) Array V, (f) Array VI.For each array, orbits were obtained with varying inlet ow velocity U in , which is labeled at the left-bottom of the corresponding orbit.



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vibrates in an organized, periodic manner above thecritical velocity, compared to the random motion belowthe critical velocity. With this result, the onset of insta-bility might be indicated as the streamwise and cross-stream responses have a certain phase shift. As thevelocity far beyond the critical velocity, for instance

U in = 0.74m/s shown in the gure, the orbits tend tobe a vertical ellipse, illustrating that the streamwiseand cross-stream responses have a 90 phase shift.

The effects of the neighboring cylinders on the cylin-der vibration can also be discussed by examining theorbits. With one upstream cylinder, the orbits of thecylinder in Array I are elliptical shape as shown in Figs.11(a). With two or three upstream cylinders ( Figs. 11 (b)and (c)), the streamwise vibration of the cylinder is obvi-ously suppressed, while the cross-stream vibration isamplied especially in Array III ( Fig. 11 (c)). With onedownstream cylinder added, the orbits maintain in ovalshape above the critical velocity as shown in Fig. 11 (d)(U in = 0.550.68 m/s). However, at higher velocity(U in = 0.74 m/s), the cylinder vibration becomes rela-tively wildin oval shape generally but with certaindeviation cycle by cycle. The large deviations cycle bycycle of the orbit illustrate that the down stream cylinderhas already disturbed the original periodicity of thestreamwise and cross-stream responses. With moredownstream cylinders, the effect could happen at lowerow velocity; 0.68m/s in Fig. 11 (e) for Array V and0.64m/s in Fig. 11 (f) for Array VI. The disturbance bythe downstream cylinders accounts for the suppressionof the vibration amplitude in Arrays IV, V, and VI,

compared to in Array III.

3.3. Effects of the natural frequency

As mentioned in Section 3.1, the serious vibration of the monitored cylinder in supercritical conditions is con-sidered as a result of resonance with the surroundingcylinders of the same natural frequency through the cou-pling of the uid ow. It is of interest to examine the re-sponse of the monitored cylinder to the cross ow whenits natural frequency ( f n ) is not the same as that of thesurrounding cylinders ( f s ). It is rst noted that no hyster-esis was detected for the case of f n 5 f s in the experimen-tal range, following the same procedure for hysteresischeck as described in the case of f n = f s (Fig. 8 , for exam-ple). The vibration amplitudes of the cylinder are shownin Fig. 12 for the cases of f n = 3/4 f s and 1/2 f s , and f n = f sfor comparison. It is shown that the critical velocities inthe cases of f n 5 f s are approximately the same as in thecase of f n = f s . In other words, the difference of the nat-ural frequency is not a major factor on determining thecritical velocity, if uid elastic instability occurstheremay be no uid elastic instability if the two natural fre-quencies, f n and f s , are far away. Once the ow velocityis above the critical value, the vibration amplitude of the

cylinder, however, depends on its natural frequency; i.e.how close the two natural frequencies, and also thevalue of the natural frequency that is an indication of stiffness of the cylinder. Regarding the effect of the cyl-inder stiffness on the cylinder vibration, comparing thevibration amplitudes for the cases of f n = 3/4 f s and f n = f s shown in Fig. 12 , the monitored cylinder with

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-202468

101214

1618202224

f n=f s

f n=3/4 f s

f n=1/2 f s

A r m s /

D ( % )

Uin(m/s)

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-202468

1012141618202224

f n =f s

f n =3/4 f s f n =1/2 f s

A r m s /

D ( % )

Uin(m/s)

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-202468

1012141618202224

f n =f s

f n =3/4 f s

f n =1/2 f s

A r m s /

D ( % )

Uin(m/s)

Flow

Flow

Flow

(a)

(b)

(c)

Fig. 12. Vibration amplitudes of the monitored cylinder with f n = f s ,3/4 f s and 1/2 f s in the X - and Y -directions. (a) Cylinder array I,(b) Cylinder array III, (c) Cylinder array VI.



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lower structural stiffness ( f n = 3/4 f s ) has a larger vibra-tion amplitude when the ow velocity is just above thecritical value. It is noted that at the high frequency f n = 5/4 f s or beyond, the cylinder exhibited no obviousvibration in the high velocity range of the experiment,likely due to the more stiffness of cylinder structure.

When f n = 1/2 f s , though a sub-harmonics of f s butdistant away from the f s , the vibration amplitudes abovethe critical velocity are not so large as in the other twocases. This could be a result of less degree of resonancewhen the natural frequency f n deviates from the f s . Espe-cially, no obvious uid elastic instability occurs in thecase of the two-cylinder array as shown in Fig. 12 (a)with f n = 1/2 f s .

The degree of resonance can be indicated by the char-acteristics of the orbits of the cylinder vibrations. Asshown in Fig. 13 are the orbital paths of the monitoredcylinder at the different natural frequencies in the seven-cylinder array below, and above the critical velocity. Thecylinders at f n = f s , and 3/4 f s vibrate in a relativelyorganized orbit beyond the critical velocity. However,the cylinder at f n = 1/2 f s , vibrates randomly, especiallyat high ow velocity. The randomness can be considered

as an indication of lack of resonance when f n and f s arefar away, and thus an organized vibration cannot beproduced. For the frequency content of the orbits, spec-tral data are further examined.

3.4. Spectrum of cylinder vibration

When the inlet velocities are below the critical value,the spectra of the monitored cylinder are broadband, asshown in Fig. 14 for the case of Array III withU in = 0.47m/s. The broadband spectra reect the ran-dom nature of the cylinder vibrations in the low velocityrange.

Flow

U in=0.28 m/s

(a)

0.55 0.60 0.74

25% D

25% D

Uin

=0.28 m/s

(b)

0.55 0.60 0.66

Uin

=0.28 m/s

(c)

0.55 0.60 0.74

Fig. 13. Orbits of the monitored cylinder in the seven-cylinder array with different natural frequencies. (a) f n = f s , (b) f n = 3/4 f s , (c) f n = 1/2 f s . Foreach natural frequency, there are four orbits corresponding to four different inlet velocities.

Ay /D(%)

f(Hz)100 20 30 40 50 60 70 80 90 100

1E-4

1E-3

0.01

0.1

1

Fig. 14. Amplitude spectrum of the monitored cylinder in the four-cylinder array, U in = 0.47m/s.



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Above the critical velocity, the spectra of the moni-tored cylinder exhibit various spectral peaks dependingon its natural frequency. Figs. 1517 are the spectra of the cylinder vibrations in Arrays I, III, and VI, respec-tively. In the experiment, the natural frequency of thecylinder in each test array was adjusted at f n = f s , 3/4 f s

and 1/2 f s . Only the amplitude spectra of the cylindervibration in the Y -direction are presented here, sinceunder the same test condition the streamwise spectrumis similar in nature to the corresponding cross-streamspectrum; namely, both the spectra show either broad-band or line-dominated. In the latter case, the spectralpeaks for the X - and the Y -directions occur at the samefrequencies, though the spectral peaks may have differ-ent magnitudes.

In the case of f n = f s , as shown in Figs. 15(a,b) 17(a,b) for cases of different inlet velocities and cylinderarrays, a clear spectral peak at the natural frequency f nis observed when the ow velocity is around the criticalvelocity ( U in = 0.53m/s). As the U in increases, more

spectral peaks at harmonics nf n appear, suggesting anorganized oscillatory behavior.

In the case of f n = 3/4 f s , as shown in Figs. 15(c,d) 17(c,d) , spectral peak at f n is observed when the owvelocity is around the critical value, U in = 0.53m/s. Asthe velocity increases, besides the f n -peak, more spectral

peaks appear at f s , and at the harmonics of f n and f s(Figs. 15(d)17(d) ). It is thus evident that the vibrationof the monitored cylinder has two major componentsat f n and f s above the critical velocity. The former com-ponent is associated with the natural frequency of themonitored cylinder, and the latter with the natural fre-quency of the surrounding cylinders. It is therefore illus-trated that the cylinder, induced by the uid ow,vibrates initially only at its natural frequency aroundthe onset of uid elastic instability. As the uid velocityis increased, the cylinder vibrations have the frequencycomponent of the surrounding cylinders through thecoupling of the uid ow.

0 10 20 30 40 50 60 70 80 90 100

1E-3

0.01

0.1

1

10

f n

f n

f s

(a)

(b)

(c)

(d)

(e)

(f)

Flow

Ay /D(%)

f sf n

f(Hz)

Fig. 15. Amplitude spectrum of the monitored cylinder with thenatural frequency f n in the two-cylinder array in cross ow withvelocity U in . (a) f n = f s , U in = 0.53m/s, (b) f n = f s , U in = 0.74m/s, (c) f n = 3/4 f s , U in = 0.53m/s, (d) f n = 3/4 f s , U in = 0.74m/s, (e) f n = 1/2 f s ,U in = 0.53m/s, (f) f n = 1/2 f s , U in = 0.74m/s.

0 10 20 30 40 50 60 70 80 90 1001E-4

1E-3

0.01

0.1

1

10

Flow

f n

f n

f n

f s

(a)

(b)

(c)

(d)

(e)

(f)

Ay /D(%)

f(Hz)

Fig. 16. Amplitude spectrum of the monitored cylinder with thenatural frequency f n in the four-cylinder array in cross ow withvelocity U in . (a) f n = f s , U in = 0.53m/s, (b) f n = f s , U in = 0.62m/s, (c) f n = 3/4 f s , U in = 0.53m/s, (d) f n = 3/4 f s , U in = 0.62m/s, (e) f n = 1/2 f s ,U in = 0.53m/s, (f) f n = 1/2 f s , U in = 0.62m/s.



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Particularly, for the case of the seven-cylinder array,the spectra have the frequency component f s in additionto f n , even when the uid velocity is just slightly abovethe critical value, U in = 0.53 m/s shown in Fig. 17 (c).The enhancement of the f s -vibration is related to themore surrounding cylinders downstream in the seven-cylinder array compared to the four-cylinder array. Asthe U in increases, more spectral peaks are observed atthe harmonics of f n and f s (Fig. 17 (d)). The line-domi-nated spectra suggest an organized oscillatory behavior,which is consistent with the observation of the orbits al-ready shown in Fig. 13 (b).

In the case of f n = 1/2 f s shown in Figs. 15(e,f)17(e,f) ,it is illustrated again that the monitored cylinder vibratesinitially at its natural frequency f n around the onset of uid elastic instability, and the frequency f s as well be-yond the critical velocity. However, at high velocity suchas U = 0.74m/s, no obvious spectral peaks at the har-monics of f n and f s are observed ( Figs. 15(f)17(f) ). It is

indicated that the resonance among the cylinders is notso well when the two natural frequencies of the moni-tored cylinder and the surrounding cylinders are notclose enough. Consequently, the cylinder vibrates rela-tively randomly, as already shown in Fig. 13 (c).

4. Conclusions

The cross-ow vibration of a circular cylinder (themonitored cylinder) surrounded by one to six identicalcylinders elastically mounted in rotated triangular pat-tern is investigated. Analysis of the vibration amplitudereveals that the upstream cylinders have signicantinuence on the amplitude response of the monitoredcylinder, either promote the instability by forming agap ow impinging on the monitored cylinder, or en-hance the cylinder vibration through better uid cou-pling with more upstream cylinders. The downstreamneighboring cylinders could suppress the vibrationamplitude while the number of the downstream cylin-ders has little effect on the amplitude response for thepresent conguration.

By the orbits of the cylinder motion, it is illustratedthat around the critical velocity, the streamwise andcross-stream responses have the same frequency butwith a phase shift. The phase shift tends to be 90 asthe ow velocity is increased. With two or three up-stream cylinders, the streamwise vibration of the cylin-der is suppressed, while the cross-stream vibration isamplied, compared to the vibration of the cylinder with

only upstream cylinder. With downstream cylindersadded in addition to three upstream cylinders, theorganized periodicity of the cylinder motion is disturbedand hence the vibration amplitude is suppressed.

The discrepancy of the natural frequencies betweenthe monitored cylinder and the surrounding cylindersis not a factor on determining the critical velocity. How-ever, the vibration amplitude of the cylinder above thecritical velocity depends on the value of its natural fre-quency, and the discrepancy of the natural frequencies.The monitored cylinder at high natural frequency f n P 5/4 f s has no obvious vibration due to the cylinderstiffness. Large discrepancy of the natural frequencies re-duces the vibration amplitude due to lack of resonanceamong the cylinders.

Spectral data show that the cylinder, induced by theuid ow, vibrates initially only at its natural frequencyaround the onset of uid elastic instability. As the uidvelocity is increased, the cylinder vibrations include thefrequency component of the surrounding cylindersthrough the coupling of the uid ow. If both the naturefrequencies are close, for instances f n = 3/4 f s , and f n = f s ,the cylinder exhibits a very organized oscillatory behav-ior, evidenced by the line-dominated spectra with spec-tral peaks at the harmonics of f n and f s .

0 10 20 30 40 50 60 70 80 90 1001E-4

1E-3

0.01

0. 1

1

10

Flow

f n

f sf n

f nf s

(a)

(b)

(c)

(d)

(e)

(f)

Ay /D(%)

f(Hz)

Fig. 17. Amplitude spectrum of the monitored cylinder with thenatural frequency f n in the seven-cylinder array in cross ow withvelocity U in . (a) f n = f s , U in = 0.53m/s, (b) f n = f s , U in = 0.74m/s, (c) f n = 3/4 f s , U in = 0.53m/s, (d) f n = 3/4 f s , U in = 0.74m/s, (e) f n = 1/2 f s ,U in = 0.53m/s, (f) f n = 1/2 f s , U in = 0.74m/s.



14/14

Acknowledgement

This work was supported by the National ScienceCouncil, Taiwan, R.O.C. Thanks are due to Prof. C.C.Cheng for use of his computer facility.

References

[1] R.D. Blevins, Flow-Induced Vibration, Van Nostrand Reinhold,New York, 1990.

[2] M. Matsumoto, N. Shiraishi, H. Shirato, Aerodynamic instabil-ities of twin circular cylinders, Journal of Wind Engineering andIndustrial Aerodynamics 33 (12) (1990) 91100.

[3] B. Dielen, H. Ruscheweyh, Mechanism of interference gallopingof two identical circular cylinders in cross ow, Journal of WindEngineering and Industrial Aerodynamics 5455 (1995) 289300.

[4] M.J. Pettigrew, C.E. Taylor, Fluid elastic instability of heatexchanger tube bundles. Review and design recommendations,

Journal of Pressure Vessel Technology, Transactions of theASME 113 (2) (1991) 242256.

[5] K. Schroeder, H. Gelbe, Two- and three-dimensionalCFD-simulation of ow-induced vibration excitation in tubebundles, Chemical Engineering and Processing 38 (46) (1999)621629.

[6] H. Tanaka, S. Takahara, Fluid elastic vibration of tube arrayin cross ow, Journal of Sound and Vibration 77 (1) (1981)1937.

[7] H. Tanaka, K. Tanaka, F. Shimizu, Fluid elastic analysis of tubebundle vibration in cross-ow, Journal of Fluids and Structures 16(1) (2002) 93112.

[8] R.J. Moffat, Describing the uncertainties in experimental results,Experimental Thermal and Fluid Science 1 (1988) 317.

[9] V. Kassera, K. Strohmeier, Experimental determination of tubebundle vibrations induced by cross-ow, in: M.K. Au-Yang (Ed.),Proceedings ASME Symposium on Flow-Induced Vibration, NewYork, PVP-273, ASME, New York, 1994, pp. 9197.

[10] D.S. Weaver, M. El-Kashlan, On the number of tube rowsrequired to study cross-ow induced vibrations in tube banks,Journal of Sound and Vibration 75 (1981) 265273.


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